1 / | 1 On the Irrationality of, | ------- dx, for a from 1 to , 40 | 1 + x/a / 0 By Shalosh B. Ekhad In this computer-generated book, accompanying the article by Doron Zeilberge\ r and Wadim Zudilin "Towards Automatic Discovery of Irrationailty Proofs and Irrationality Meas\ ures." I will automatically prove irrationality, and establish irrationality measu\ res, for the constant 1 / | 1 | ------- dx | 1 + x/a / 0 for a from 1 to, 40 I will also state the cases where we were unable to do it. ----------------------------------------------- 1 / | 1 Proposition Number, 1, : Let c be the constant, | ----- dx, | 1 + x / 0 that happens to be equal to, ln(2), alias, 0.69314718055994530942 1/2 2 ln(3 + 2 2 ) Then c is irrational, and has irrationality measure at most, ------------------, 1/2 ln(3 + 2 2 ) - 1 that equals, 4.6221008324542313340 Proof: Consider 1 / n n | x (1 - x) E(n) = | -------------- dx | (n + 1) / (1 + x) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y ), it is readily seen that C E(n) <= --------------------------- / 1/2 \n | 2 | |- ----------------------| | 1/2 1/2 | \ (2 - 1) (-2 + 2 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 3 (2 n - 1) E(n - 1) (n - 1) E(n - 2) E(n) = -------------------- - ---------------- n n Subject to the initial conditions E(0) = c, E(1) = -2 + 3 c and in Maple format E(n) = 3*(2*n-1)/n*E(n-1)-(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2+3*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 3 (2 n - 1) B(n - 1) (n - 1) B(n - 2) B(n) = -------------------- - ---------------- n n subject to the intial conditions B(0) = 1, B(1) = 3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 3*(2*n-1)/n*B(n-1)-(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 3 and 3 (2 n - 1) A(n - 1) (n - 1) A(n - 2) A(n) = -------------------- - ---------------- n n subject to the intial conditions A(0) = 0, A(1) = -2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 3*(2*n-1)/n*A(n-1)-(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 5791521161186858631080408209799374831889462098474684342671075205254265385407\ 434001427223602040562623298809918867545316973280895354047199418781451191\ 411894459634356350021171137225411394901338074515078851908037847923014123\ / 034072370930228201 / 8355398858447771845486672184221321266331790135967\ / 161139740582913441936390627567694641780232443985906202095883486834367616\ 318670283425463353826251116748466118668815006942113707417197588298513630\ 203016421514931322593164640153865755072888000 and its differene from c is 0.64789684392576917390908250866978209095576422273695051804704888881303065870\ 036957879183020539200160463696768413978655912404742634229700633919270816\ 742633521335134871770904910980161217097227473700056668302696630269539648\ 990463441802204371043711553955827735222360867984990589014758328089947181\ 830746625416168958998683970187807146876163929700172928930344459294375695\ 311339103067401935143755358674381185003355960644005796577806455071236057\ 633017534210032770015377462808928207435520528082836979654795984459765440\ 125697485828009977974768972820800431618220207245390746922868722615837281\ 648279256623099306714224448621737301079064061568443456452809053891013777\ 627988817463080271643719504203070459293554602255007287553849005090611402\ 350475324589124364314618457810268630152303222401217196696553331333700757\ 314694602337467484768147469858939238930565477241651691475466334388297497\ 761393294440118891928890074070742147550490802228074781921523220837630008\ 150549721562624257946085023720493026445835655357559208223672387326950277\ 619608183474007078772542703746466601655186558302150990942935145651269897\ 940517516169834617995538914906927308441517591510708732567203009505010870\ 631727556792138011872517101254399198543821082065521161207551595938318941\ 942929829780097817951038356276603324638921495607556015765960057694441525\ 565182706931771108967176670039847859602636074140353987538092612551740710\ 422975428186196645892749155493334317759462103076389141242338344860193072\ 463733444719280963858842301313854056709867171886503261126347796284182843\ 968017343767952364620173231307431451854511805032551657156350568754485888\ 444372503971033043698268648000017475887512005924753426272509028772214531\ 820380011517383774244555424218229125554205007560851594043126043608333964\ 894380619438010303445375963370009223935963267816247101507638999032537011\ 460485631467839367862462574908795551215837490351242883148079080667867145\ 401299427510977686302080597653380684601824338860094193760656459196790315\ 650697858694477578174255641195850224207765124133369130334001317581041858\ 202454860428531084004793430303261346577123665331429625867359222934226210\ 305975581830270747712301820787223278945705632837223445886971618111871057\ 389464531443974544787572275826057289535753970760302602103850831155724782\ 601447645181360603568033141296386296532190710074474685969869356104890975\ 063863609892829121248710103055401955052853514873376291238891621079652328\ 460767209117769089739067058817276563614245433385359163170253540470422700\ 987057380741990040667100305909114790181935350512136432410496839082031240\ 983044252711451832167325787889722315678723624396461577582956583577492864\ 484143052923572613886624275115345615118364232718404973889593385460838401\ 133711523128433412250788269629221846358317389177998274764778721175475652\ 032111282293712958058116468421275733204740627562096652099005097891542846\ 266732997567182721081087543845207889672337394447463655498798634415935701\ 404836588596749222135093312871304746148103326467553643872483830813312287\ 791634873628032677527945314567306286529606036784120770559320612066299560\ 596733204137671903285445146841962244921527415641555593477791714844749074\ 390889337281920712539333183968736554420638105971460558321974454950750747\ 384054969408470892981119659538780976569094184063403128723923017405329253\ 944393288492327594281686136268747647278847672876911891319309422408881871\ 804387400276077627381375517334162391399669124779856108689911518374764325\ 734644154231182637671466839642486922711659161200454719524252692451589056\ 986573689714567226194532875723043152500831665494121156617908135759643273\ 075402142801659963529911915310828997715838566872437284443265855504502269\ 440447645816169809148275887553373477657804147876158802880056212453532627\ 343925974215481259650512191593922124186581812815605856937223055238771147\ 697991138903657041845884948843899895024295017249528924004462880157862971\ 200395150914908032847494658220979789828342162092038147075975693885165692\ 072609917184891693364195722520324366057001436928591170692501088584544476\ 071665799341214025576608695223354777795680841026050284235917870953041797\ 827005442419046857598686318822727272916049139217654615952930318583963912\ 506776884740566439933126988794199883858053162652705371412696458169632600\ 044591083086984219172331944802975371458719078776256740623237008515212161\ 258318115534334768678337073761214249933531532555322677941149549515306021\ 128103145070340005781155735946545380985034134786325053757707899309122687\ 754259631719170029493237271221030201700795538423784114751350189947555514\ 793526843193043166639964501988976568552844396583095970441586027875466528\ 018956400309756210297721435566833802192046444732790994870193298459274492\ 871591114861264240209407865880931133679453790762386192014187620429203883\ 038520435533149165885740139076414519023001597893173340480778276987955594\ 931876732842947038527112563515026228665970757876993272986436688114075789\ 362826854539088007898668849699804716185955269397590613016570308574832793\ 944140534872009837065473400005071008519611047131757117401540170092438969\ 029748601415043444459089603797032989655787967866624625070067246184431845\ 756208857252237693465874841246215732555401720721951425181261880807618971\ 849403224325456462492716703832223073384727077648089396682535277212714704\ 057005840651685863254027338004058441653776307176620195995701652451979679\ 259007454632228142568531146587391282816655422049105415307829955403327070\ 492776175967236922521703245415803684930884942634534401389883072223429105\ 751767893019496645071833865378017887109135995739450430444910321450429610\ 400431860126024361504562407175448183760909509666012514695950911062579334\ 528754297278258994707900955334815109455063182021215077212578748953276577\ 512315111127752496472956093804020225355083789895848510981732925665693153\ 260841330648778345859176941385288062914562804325789641608176684269661968\ 484731127440910155069863294061493759558732244043551000621521671743951377\ 265647033970193573057480044304870214465994399654979546915578147477916119\ 147211854681830578744909313561383376989126989975322370262044530953685381\ 230833484274596186211337190961914130336279241830134397453773776036602471\ 221504577522442606442868091924059104096380947814861163145418722568640693\ 036709768739680959648941618892371488484611565364932385391897931518418746\ 445313419009134746265775052291676518496167157254944541961593575150492891\ 983309591682067594487074254437217266942926250768148516628565249225651350\ 290912157737251486201278916899354913799672937352319575819678078205126913\ 713050746375960440500974296837061295999809067947840717645485765827929767\ 218612374724966022404074271945067600204845568273248843454419838619171542\ 346312476562387750955107658763285837082589457296313387323445668171922018\ -306 74412985041780604777833987518031956150239538098641780048663153198979 10 The smallest empirical delta from, 100, to , 200, is 0.2869282217 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n Lemma: , A(n) d(n), and , B(n) d(n), are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (3 + 2 2 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------------- | B(n) | / 1/2 1/2 \n | 2 (3 + 2 2 ) | |- ----------------------| | 1/2 1/2 | \ (2 - 1) (-2 + 2 )/ Hence | A1(n) | C | c - ----- | <= --------------------------- | B1(n) | / 1/2 1/2 \n | 2 (3 + 2 2 ) | |- ----------------------| | 1/2 1/2 | \ (2 - 1) (-2 + 2 )/ But , B1(n) = B(n) d(n), hence 1/2 n B1(n), is of the order , (exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 2 ln(3 + 2 2 ) + ln(- ----------------------) 1/2 1/2 (2 - 1) (-2 + 2 ) where delta equals, --------------------------------------------- - 1 1/2 ln(3 + 2 2 ) + 1 That in floating-point is, 0.276082872 It follows that an irrationality measure for c is 1/2 2 ln(3 + 2 2 ) ------------------ 1/2 ln(3 + 2 2 ) - 1 that equals, approximately 4.622100831 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 2, : Let c be the constant, | ------- dx, | 1 + x/2 / 0 that happens to be equal to, 2 ln(3) - 2 ln(2), alias, 0.8109302162163287640 1/2 2 ln(5 + 2 6 ) Then c is irrational, and has irrationality measure at most, ------------------, 1/2 ln(5 + 2 6 ) - 1 that equals, 3.5474705913691086065 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/2) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/2 ), it is readily seen that C E(n) <= ------------------------------ / 1/2 \n | 6 | |- -------------------------| | 1/2 1/2 | \ 2 (-2 + 6 ) (-3 + 6 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 10 (2 n - 1) E(n - 1) 4 (n - 1) E(n - 2) E(n) = --------------------- - ------------------ n n Subject to the initial conditions E(0) = c, E(1) = -8 + 10 c and in Maple format E(n) = 10*(2*n-1)/n*E(n-1)-4*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -8+10*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 10 (2 n - 1) B(n - 1) 4 (n - 1) B(n - 2) B(n) = --------------------- - ------------------ n n subject to the intial conditions B(0) = 1, B(1) = 10 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 10*(2*n-1)/n*B(n-1)-4*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 10 and 10 (2 n - 1) A(n - 1) 4 (n - 1) A(n - 2) A(n) = --------------------- - ------------------ n n subject to the intial conditions A(0) = 0, A(1) = -8 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 10*(2*n-1)/n*A(n-1)-4*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -8 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2010659012236817632567720263062203868247817836999695200378898159353537408035\ 967411662936670773025728423264339970273048069452334212163807424082502432\ 405167886245408933496517918619700646696714304332206223149892565508123549\ / 88807292443465798961791958990549443707214265990815617136128321 / 24794\ / 476417690197265138583887190823056747046565995740226265080159957907557029\ 053012948369239029184935284911798085565302503853065368591981350165254647\ 041660878390072679742533933377385428940047532148545189750378578721453033\ 7729361741764986772007930404810305427963399114143829097912480 and its differene from c is 0.73603262836191417545712683259365593450919474736702487443353621419578524873\ 105092215886015766749583911076718083343524161437747400178996715528717782\ 309815387794633929897549483048052151783429097820871362241665330366739226\ 282402032607915213703796142959546976787970929585433370750455315015234998\ 753021189492982728589239787993083771838323673580901430693700337965701840\ 222394797901981852463948384345422804967511183954526338516714446430172954\ 575247352981688495829805141286541456710001335751617010077384535347103147\ 635309290430500682179544652461332255806976817543067574789860882360575280\ 301995318578393755570711136702857831422482314247832905722797717387402174\ 506685133291001324975561426134228247706252362759595312321709531427588788\ 064869492159975778594168606797693451592012146900008127943276157807197791\ 534338574986670918711162096003416936637464443098754359615752215366376692\ 203083523490907175518595207388618014155049247068963637793505832425200812\ 878225281309774562382473606357719933942987481773853520678520455444064410\ 230691555815874485069178241971117712001190885448759553386505947684689147\ 912303083237791360325993066915035176336457604634016996930451384906210270\ 644801304513982769709429716215359221923159375141829881448470850056993618\ 879338890351922300909853008965880718793839324353627464900700468685412978\ 361772379950830809974115655890110370318983499491414506982173049209774925\ 458798818199013740075546862923472362607534023350581806317202170423692528\ 345946937629696961433230229286961166327916586260143199110727628123594722\ 503969197989467845992033909832735878516216769238576166411047852046647479\ 692645357868656576086252500129716418880427151629296617977540059461795520\ 291764862946513047632782632350834130478789411193020524979496013198382912\ 657008357171624625980955716647920677380284114604757107334279517989460891\ 764979061501968219111884629174174247886657267482858991026122530811529464\ 242016183283064221006672189898747679451071910345783985999641054738378796\ 258054535875321402951266616972223540295844178777039626257490025020714637\ 807160233688891706480560909469630818262869439209324445003916191058893660\ 298236582825681491847384604444773911624975554595974522700224330222480138\ 649568031292856800422378604716253963703334808002193898931210497559613207\ 830604406664662521294109374388315829870507462332221241614162897952885180\ 879152880788044734732000362781157446416028873286377348675084237967272302\ 608057040695338852380447118463566766856820981349169106566090274771481269\ 877406723314042746955637134151654496284437439168184989178428751122691504\ 960153705326002642816930918935647362945278600143270936624193844761924625\ 052760688246430763731504410399755452705197016954648995230278023992067236\ 636606266002984834787731039510086960247268753432724351641914349389871529\ 591854936739717311096363978654900149882932411678028712877809979885119397\ 282078874821150619035992923362627610240584020382119905780389566279087729\ 567293538022402643574257654378411009680392788874439032666244454588387292\ 075542036739922563909864622403707798881256987166679887629061856449226569\ 885873753994200295679041788640515620046526343032495595917476923955418018\ 626347468899891614334287730249438167535518444363461068921136509474654375\ 209525999107359005467222027910120125590747084990879720874310707630029379\ 712904891241012667636530691847258853285772640803698457481160636739031292\ 895870298655420783270321220911551947947368493626336292124969765298150252\ 878108172769475573820735802637539709698437338950912007979619106263602078\ 675153851708076946607738261556194676310995234682195721254412640080722380\ 329866424568438977755768536354489916668211915611113013073284880762388735\ 075952459469935783289428926904107262554049973962571504116178263755286474\ 880716040231171831937857201108077085336424228680002580666726260674885516\ 283518751902886010787101048878279079323007927413547446689847832311699242\ 667754758313257703902424467060638406882131284421262542043657409162494321\ 034508332168211380270696694974214636940898743887991415015677167963398818\ 906418774753375901823606259604429243624681799159912899915797604309119799\ 624711255442428146589081607340004548529869622536706137014380479866625068\ 176897459742096000366866350551115356267824868388243180368089474632041552\ 005443927648395455375923672644727253044935840708255790349932229183836421\ 012255124653754262842075737945591087511434454036244133535136124408990301\ 732359682591785492204058220988188342640499614127822118636957524471900112\ 357905353864286167556896765811007829484791027635691756717918349960981277\ 614368149333845978836991906903311375361988617466495127288317902587406371\ 960404340540531389427231412812007804409668749570011282234462007648263866\ 588365906214067969577383177594590116084118834017744876106327962573461615\ 121624470054478872500704214073140547670198406177008338741004725320219070\ 890542626927843628286635114174988799625015454399823803351392683367806519\ 787610310736991661646302809963781725348878449822990236493061171912757425\ 708322340699442114503123669979874055476002293091103655190695666004481480\ 712791275181293061757636544016306305295954875470555416064741533641958609\ 925491721147561595116989440543789670516601590768094233527668117657456173\ 274399879326497337662513271930794559782663399324136487123491087476015625\ 423554135457053577639133290836342091002028779739124305411078132308808643\ 519062111278968283557136990177689286924598863411185455372655193429905666\ 794875251212719517086191457949801382635269637969879330718296678214739837\ 341896722163193725968912234575687676885872598211686215977615062265813002\ 121658746199713081544272588749226408889784979274940077729813354942980221\ 902286229856818406974792702499299519865310289086126877802931249740930563\ 831391086801315905226089740646911506517885986042622030404210444172649200\ 073053225027819246726179459790834876619196934327441833699305774923177138\ 846972613988237368186391435897304423409398740057494500452301569020068095\ 249317856361290523178122156059249178871988081956623553528674460446701945\ 915204399822795089231678901722528044146204081124782681578342153639795464\ 177310512848208758244980360300231965050719300861153537609699678256081860\ 272013315532429847457818676572765958989944750989121792532436313773501409\ 163537000102735182563006791493467989834849361167608931893812385040548575\ 878567325554678562523000089490001493552896039257345035984991121652632129\ 276483949630038252878709971023974704269515117571366068227190759806296547\ 764064500965708193152742341745143429061048810244611454746528412567996852\ 263656105841979832558085464113091536438777892529256116449408304068039569\ 790636377391375865351241027103421246364265430929865286764318656321946279\ -398 044786685454271461473077475846640049170263342835 10 The smallest empirical delta from, 100, to , 200, is 0.4071733525 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 2 2 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (10 + 4 6 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------ | B(n) | / 1/2 1/2 \n | 6 (3 + 2 2 ) | |- -------------------------| | 1/2 1/2 | \ 2 (-2 + 6 ) (-3 + 6 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------ | B1(n) | / 1/2 1/2 \n | 6 (3 + 2 2 ) | |- -------------------------| | 1/2 1/2 | \ 2 (-2 + 6 ) (-3 + 6 )/ B(n) d(n) But , B1(n) = ---------, hence n 2 1/2 n B1(n), is of the order , (2 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 6 ln(10 + 4 6 ) + ln(- -------------------------) 1/2 1/2 2 (-2 + 6 ) (-3 + 6 ) where delta equals, ------------------------------------------------- - 1 1/2 ln(10 + 4 6 ) + 1 - ln(2) That in floating-point is, 0.392546239 It follows that an irrationality measure for c is 1/2 2 ln(5 + 2 6 ) ------------------ 1/2 ln(5 + 2 6 ) - 1 that equals, approximately 3.547470592 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 3, : Let c be the constant, | ------- dx, | 1 + x/3 / 0 that happens to be equal to, 6 ln(2) - 3 ln(3), alias, 0.8630462173553427823 1/2 2 ln(7 + 4 3 ) Then c is irrational, and has irrationality measure at most, ------------------, 1/2 ln(7 + 4 3 ) - 1 that equals, 3.2240532881366202828 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/3) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/3 ), it is readily seen that C E(n) <= ---------------- / 1/2 \n | 3 | |-------------| | 1/2| \-36 + 21 3 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 21 (2 n - 1) E(n - 1) 9 (n - 1) E(n - 2) E(n) = --------------------- - ------------------ n n Subject to the initial conditions E(0) = c, E(1) = -18 + 21 c and in Maple format E(n) = 21*(2*n-1)/n*E(n-1)-9*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -18+21*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 21 (2 n - 1) B(n - 1) 9 (n - 1) B(n - 2) B(n) = --------------------- - ------------------ n n subject to the intial conditions B(0) = 1, B(1) = 21 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 21*(2*n-1)/n*B(n-1)-9*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 21 and 21 (2 n - 1) A(n - 1) 9 (n - 1) A(n - 2) A(n) = --------------------- - ------------------ n n subject to the intial conditions A(0) = 0, A(1) = -18 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 21*(2*n-1)/n*A(n-1)-9*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -18 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2584397855179401134639223963675126206344920224235993695962311008200141179937\ 777117109264280701801233879002497242239531864271596068792590615186789807\ 912377816715155193034910103693427595550385409200573307753075713744557151\ 102313497017500160785585573378177215922345817539563514047111852450965813\ / 0285984258500500179 / 299450690265330798637979336542332311435273433397\ / 602364823309612632559600002436490881032680627658729579690558935695252003\ 522385014180359674061118849475577389567787852795835287138468749541846052\ 051405931888717497437263466587245743176792917378675712330593105129849191\ 80873989774853228542877622460474825384789816000 and its differene from c is 0.37395631859683921692601937483802289468536863382052652946734581158061425355\ 140881421654004925603170733884450548042076507022822084403811184094363570\ 550397013896421774862537702568782004810547397469697398598284187302848537\ 361374255260576086827620450645823230595089201790045634671703114734416088\ 594616586019935135795429725124807886219318200521977577282038958772629733\ 951667157023008448600500913177555833763272065035170923550320864229893983\ 493638901613419551902526618401525531796335229816499091812392595339070388\ 918491512094331574028397667599321046217455645165569845133246041152085834\ 765254081844297674723012716231478445454783904744240783982899881950887931\ 241709636958057969489577637003180187199774493279948897886358663038268861\ 067336223198013180407956697903967783478790255235011596471711072357339353\ 179460991910097853577956944566269875624181909409926841109763657387490463\ 244288291562878913529281002661803375456740241733726773548561336470455024\ 888846470557875935539638078659577999663146668497199139326471109653924717\ 170984837900671288525844975242562485659626839373841498163019265232021013\ 260968264848790054090944320353032893428408083126343381510499298332214828\ 910243034644248012723746553534560213730630255874716602410532442782176876\ 028397145856079194662284141568762557626860474925064767527471990756530048\ 113824152911239387697044524054845122615314202115144161934965058668349549\ 440079422012362347018565704045184803759720434461775830620862505706238950\ 602384977855685100935924625438002007366257461169670277454119128496295733\ 696404561403699498947496591218524911663963124534964122505435963865374034\ 724356963331907706985098473143182562365143778342163598881190268131433354\ 613584375334224018413237498216109305453518449830895403118895776225926383\ 843686620015219421264091269991113123741511747738084536767910664486044611\ 665860126977256963967443784895757340681526543387947251667558489611469515\ 367667835148980760340617779744565943364186255631893357441246498084951296\ 072431666004863872035498756815269971509266277273349706168536651046348997\ 204026247038883634627558351080714228281555230114658788273327121884606582\ 892740979915447518490900380924434022722526445814263365585649766898821557\ 228433222150098513045498601686890832615310406252239395543277575040116176\ 508180510147721241074477908954296123154140018773664215948753076759739082\ 250843354362306521348619413987791155018276816143341920329912722033714967\ 576896084244614026370323967670108372999978258791671846070739685255319063\ 427802061959463541416198704347440184138885114464287152586117074938863606\ 322384460608585819239891874481877221754941504090570705150840210945413581\ 027971186889461394773218648997629288176242474292935910089686035555141292\ 080669885436090374635937793223395578767109322862941353490319407807824801\ 388064748215019385309215955010332983095233505810648690207441962055099107\ 030636211483737309771769310117871305463269830595891198026145472937429080\ 396775331127217257315076399588421913657557098884017978116812265912137890\ 407040620158754924003824795326381155651076744365395987402771004219283373\ 090925066662539953842278451669797825562405428254306879481625074035630211\ 232612867535778276208035468764363845407831288154011428698622016741001609\ 777388342012623703057751594873679199979234408964050002984520557094739045\ 822493122948261862794304177709387964219618930493445832686359958039974779\ 991709873826654584703572503987364750518751772322050395718938605831275690\ 478205040560366351057663418501121136450914610576514147018084397635660476\ 375226629817732210333368527797824303714028992355659100086282747644071432\ 050499781351611833159162562418362248260062915219777980837149448114644401\ 101341979573586013835223927523062619522623672259151580488402726150181618\ 347990034792522498385246175908143483002283139282441456051849249047528041\ 387503920960076515089062743133436232524923652156027727567578230379397851\ 277875032268675508788456967963686252136447813115951555881831747152190061\ 830890724021759882378345739206199498910635716954906383694802393965240168\ 645755837154637356714837889425089260718181281207455314840123121287454881\ 786770753381648234352095692805298407003561567424172735220518575134668996\ 031121395314155226922150283851429868937982521426844134103986397344476218\ 627884242806082578774501061568245836184009955326793262423491020913958106\ 940995247310463524116690368321091383836234156304042626947560656510755410\ 535846363706043538407945216132523740761602900966915846284244745107561849\ 196103616055739102562062342626278008372937946514358110041212393779479872\ 407737640308633369389189505126903939453520211216184082242298868044855851\ 455784221858767925035109872834661279003299650182117170000653113789268896\ 068861840783894679231158629963380798689545832061043098338518199850993557\ 769514642656389070899321083053545205556219011552550268402651346361199757\ 563735875755051966750864018278129993104341842275275240348152696042867649\ 345460480094222352515862440393288531283430607588855141815619202994449043\ 766601010410075637397372126350467602694749052718422303074060957772721087\ 402889223005600545080196175389806129743479424021890926856860747272481938\ 193857789423959299857569932318627581233207274422833235067528980310236179\ 785278833427632639433808871301016424983075874414005051136603927102045280\ 956752475933083347101251332374772075472891089218640269368760499768050460\ 159855393034734876115848421830729667188612993365928175381662070390499524\ 282576088850921322334416044278164955799185249659176752382772045463049568\ 778704156949133454544946521528884781999338996764537373085435725986722262\ 806002957539617494854091293840773749943131268414131912088488223569687895\ 411943796039438563864054768195082670262605293621535579558209806446572467\ 977366812442046886383195354937892222863005229789858643046137816547475683\ 816821207078454714529802217876250067930818089613647733819568779977754505\ 699202413845586278485988033157628034767147950075152897952415617286098715\ 101325242967332657471216787398189785834033878397855589274706689223412853\ 899362329929776329196963549217472823450336754452163600202580961543087749\ 903510011790682722852794989008382164111859948255933705358102092083088501\ 019347166096507184522286379416401371032614128562694935594401356077506817\ 180562831143140694744482880728687373692289630893139274663596395997955709\ 779397712751870211409242915706575556859017873447749502955615858961386497\ 862221760937227093182641407890627121670745417101824334335024329321924250\ 103755367735143613307778283883347505925926354185249137426339452211967128\ 088346059648226344170115680946970171040127787204238822050938016851207806\ -457 4342761644360427632872475276118209273990690443925572953553385 10 The smallest empirical delta from, 100, to , 200, is 0.4536042110 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 3 3 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (21 + 12 3 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------- | B(n) | / 1/2 1/2 \n |3 (3 + 2 2 )| |-----------------| | 1/2 | \ -36 + 21 3 / Hence | A1(n) | C | c - ----- | <= -------------------- | B1(n) | / 1/2 1/2 \n |3 (3 + 2 2 )| |-----------------| | 1/2 | \ -36 + 21 3 / B(n) d(n) But , B1(n) = ---------, hence n 3 1/2 n B1(n), is of the order , (3 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 3 ln(21 + 12 3 ) + ln(-------------) 1/2 -36 + 21 3 where delta equals, ------------------------------------ - 1 1/2 ln(21 + 12 3 ) + 1 - ln(3) That in floating-point is, 0.449629506 It follows that an irrationality measure for c is 1/2 2 ln(7 + 4 3 ) ------------------ 1/2 ln(7 + 4 3 ) - 1 that equals, approximately 3.224053330 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 4, : Let c be the constant, | ------- dx, | 1 + x/4 / 0 that happens to be equal to, 4 ln(5) - 8 ln(2), alias, 0.8925742052568390230 1/2 2 ln(9 + 4 5 ) Then c is irrational, and has irrationality measure at most, ------------------, 1/2 ln(9 + 4 5 ) - 1 that equals, 3.0597312482455252025 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/4) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/4 ), it is readily seen that C E(n) <= ---------------- / 1/2 \n | 5 | |-------------| | 1/2| \-80 + 36 5 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 36 (2 n - 1) E(n - 1) 16 (n - 1) E(n - 2) E(n) = --------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -32 + 36 c and in Maple format E(n) = 36*(2*n-1)/n*E(n-1)-16*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -32+36*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 36 (2 n - 1) B(n - 1) 16 (n - 1) B(n - 2) B(n) = --------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 36 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 36*(2*n-1)/n*B(n-1)-16*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 36 and 36 (2 n - 1) A(n - 1) 16 (n - 1) A(n - 2) A(n) = --------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -32 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 36*(2*n-1)/n*A(n-1)-16*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -32 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1804238414206357904355523320596475287392926160246033174015920605437309060339\ 240381689233824692461811264534340496554261270470590527919697653544091604\ 859737506651491921524477257912793608942825118820857073562952758582665708\ 755594535314860883592036218727057479961758103689256095600198755826203464\ / 386252171223505188736557738517186757942857 / 2021387581648952865099567\ / 564777049483534785158894184578668379190769615169380592240687953575863620\ 020426951219594639439993871992308437729557773839743878999378067296344337\ 732967809263217589146705690456944456664709647264045103808809362032831992\ 887908688933271141073739651475547949632776251485789664189203918600675143\ 836228062731234402000 and its differene from c is 0.37621190155527201929548533661386869041826664974422439581225884585685188345\ 347435692059190510791753014399380926856242273185108801613669781287301026\ 839717446053330636996179115481668925529668022704888914373684887653899620\ 709528303987818298557826342124976091197159007107907324717322812516950802\ 732335927822545172207286905019748056940486134009326324303848081424687800\ 865057697792747589748729492999178484415035931677503281715274333130248817\ 439208144129921834846292995096844242962616539944786964771945983755258178\ 583082389791647354087551173295798883529447909749342290936470834388158932\ 275870609629838889705519476349090705138376543943408171391411952921821000\ 981280224634006230637671374270473219792594794630125484153109762764405246\ 482796184280350879024607700623378791071879405448163489087141700131696263\ 387609695355324582679894409729031261109928987480771323322743450895891404\ 254628550637484770638578591107642663566496175793980496374485587597034505\ 662609512220179423462496900936262236746288065767431533002056254366697250\ 875995723306398812163202982751207854159455791607938848428710514221290768\ 524299979104908495040165283154650231580860537389946543192926868923156719\ 113433346765114109847843281582700774193163128234431807772672045424733284\ 343881137444360513260306781713175286931624164365702047094055112147824074\ 995085538697554213969216737459382747793251005711147426967396100879244926\ 129197633535071210734582943505035438705067173496175032282254846693129344\ 171276280744590876145888863426974361560820464966300154267261153514559263\ 961684341442016715943594323056758934367391344953240457167108614439187125\ 859015661533337119014604946123290051274728512775899867452827924063887338\ 405522837844790605953628804996689698509823896655205337985617965930448393\ 263447466707029051372361549642606320533155398506428451417323062333022838\ 334515195321360022467371274787668119967924967711709559339013177466614682\ 489501186533765234442453365245229531513901005376241273083121559564357037\ 978050708858977043218520539597727699387822570013353860021029467533782884\ 686332957707742765403266247699708049720438484559070451204315948024229614\ 550979669270448423103270901267444078770025422376398007313936081511564979\ 595946614684022620250987589728128420089055543969771106384745670343039274\ 366443537586336540729589978117040221208071109729904232777532662357322203\ 650885431184594223879256939049109244656728343020537854776755676618279102\ 437799068381145905465158602943337393541920114088439980964203170803197727\ 923975503108795481782330332740905076689478967238389329949567078278456771\ 621256374989678316810840594559008572700779416999377575620108159547233986\ 712123557383982652190306259861791791715113300019444680424566950366168249\ 888279050535442294422141680046034675760767396944520952700842026686884826\ 192340479518783387288336069498848245569466911463071220633682892799706857\ 564225033679574357153621680382579258684396738104311055938313234483301977\ 692550551570241319085935485722037332675236190319498439879867294811739176\ 907499949227376817788502520148656546324667443990982127594664573812101201\ 688388630962264175595263419336496682395136322102770685071560524191665612\ 601362865515433839041469604458664124624775257051732940387350215953582267\ 508052266477284681503395311726194081606885663272672358404586711687749758\ 216111879856428765065131584078565884374653176910020985095166703715968099\ 759288112946117282579393061284418746919081689081284332177492043155954520\ 730636315695682783777161529632458744733539786913713848375757234951116942\ 153377257244518057816317023356889738067550850973507332973513316370958772\ 709130555078802970207298302133543537057720026978030658790207957934948761\ 708630953360719801683890377973878218210882549354709849457342746863886485\ 758757874628983164193152735041232303374324997483162026218265475536288883\ 165338860719221482113496354159077120430049470109423842150256725398257105\ 291302406407833049930733534829125474054048596725816132556031321594135928\ 199548044961854794899325666896534656292245979122877400403203840730657243\ 372396665699363904168471381788585559858051464114430026843046443269714002\ 710374602687112970611406809380224665395274972762870924980463970739451748\ 235462088823918189115280662227752713496965376207554012042154455796073108\ 411550316043815544284306826965355057818875195072844544522031360671087361\ 151233007474727165034904627862151771029854413294669970706385699359099433\ 446564145013454824594385085920377631979321177867805219027888850467999292\ 892950882962192535370038594882036425126198176851425968818593829200630395\ 972349788565517136446187759475388357044108211606680392580992756614676196\ 558893064237649678117979680068791618018823212415281140399160029757263727\ 143633186607222124673281224673614660351667924152858152885275377507211108\ 173227432284507156064129332533435279666284926936672183118119882399225352\ 718385142817603205146451782040746236552137512235802393105694162006705780\ 229423925130449278102707592326981392498661151440799789669378205602266689\ 464473063241673691067529398275446206938855244188904334121025519730448268\ 052394356762803964364412208837434669471494269363954643135858713502676020\ 154873277790574779638890559190886217409009304463517676960404680591204517\ 743283074656464799163762551377558151142135176917112672727624926911989954\ 197053289737275933033780453848744090983430344866780825729130655555109093\ 763925679878946184901860739189687770634693330207444854880359576392711560\ 248081434232079284671515467741807153491160411827658988255057497498295017\ 145032190334865465525897847918253221214779066172051279831938416949466252\ 475073349725501132868842578578669173443386039303673768029619998417180363\ 474002953775657374041064941752354359651467206381530797394117150782754269\ 795863464800622242389924228661968545733833001722039606147882436467323335\ 841035117139290867968560352712110598537564971774791894846637030580309668\ 371377841101198268314763060175289547855903325084855557239675754013187208\ 635216869174572251435926529293432678065299252886364006955223183146464293\ 367953123333145445480513076178078968760348075582656891127501634776057782\ 515538163048083153861864197371209696227902478936187709781167272561898497\ 622413664904845901748494953695353391325384296405630028874228981343172951\ 867962873666107737849166165062142753288138136620561522400955203649959782\ 366477771416719738302829377021138206623478365257026722983044546319275640\ 852302804801215200989488480468237327004693588410800533487328252503315355\ 516004761983269506199793097478213531589055817721727106384474716492609033\ 733411462938288728409688099035197913416877223898986501329857112546718781\ -501 62107121035606788 10 The smallest empirical delta from, 100, to , 200, is 0.4939170681 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 4 4 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (36 + 16 5 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------- | B(n) | / 1/2 1/2 \n |5 (3 + 2 2 )| |-----------------| | 1/2 | \ -80 + 36 5 / Hence | A1(n) | C | c - ----- | <= -------------------- | B1(n) | / 1/2 1/2 \n |5 (3 + 2 2 )| |-----------------| | 1/2 | \ -80 + 36 5 / B(n) d(n) But , B1(n) = ---------, hence n 4 1/2 n B1(n), is of the order , (4 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 5 ln(36 + 16 5 ) + ln(-------------) 1/2 -80 + 36 5 where delta equals, ------------------------------------ - 1 1/2 ln(36 + 16 5 ) + 1 - 2 ln(2) That in floating-point is, 0.485500243 It follows that an irrationality measure for c is 1/2 2 ln(9 + 4 5 ) ------------------ 1/2 ln(9 + 4 5 ) - 1 that equals, approximately 3.059731204 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 5, : Let c be the constant, | ------- dx, | 1 + x/5 / 0 that happens to be equal to, 5 ln(2) + 5 ln(3) - 5 ln(5), alias, 0.9116077839697731311 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(11 + 2 30 ) --------------------, that equals, 2.9574096761096127778 1/2 ln(11 + 2 30 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/5) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/5 ), it is readily seen that C E(n) <= -------------------------------- / 1/2 \n | 30 | |- ---------------------------| | 1/2 1/2 | \ 5 (-5 + 30 ) (-6 + 30 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 55 (2 n - 1) E(n - 1) 25 (n - 1) E(n - 2) E(n) = --------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -50 + 55 c and in Maple format E(n) = 55*(2*n-1)/n*E(n-1)-25*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -50+55*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 55 (2 n - 1) B(n - 1) 25 (n - 1) B(n - 2) B(n) = --------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 55 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 55*(2*n-1)/n*B(n-1)-25*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 55 and 55 (2 n - 1) A(n - 1) 25 (n - 1) A(n - 2) A(n) = --------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -50 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 55*(2*n-1)/n*A(n-1)-25*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -50 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 7573116032081828167054609943722415855328138407987204311278630057515289769610\ 798710565969853824869761451035867819370252201866990175146459650709878211\ 669329670148099646353555105841117489581459399393845384017820757252786407\ 224369064506840044663574249857300061474943634001663994328712234029887045\ / 191477526365141747705284479698364656195434932621376377675223 / 8307428\ / 002757088902319143489527329732691961089221257269913953020629290934232628\ 786048377427941664366679588471746317403500555305896610558343153151921112\ 822680222098689177944551165641139444494431933351544983130371901976572378\ 635876412350497477634346532837515891189907191088013181095752594210864442\ 786128936706707579875144166413088345670339105388441217600 and its differene from c is 0.35160071302462573559381167484714613119428686948277156712896578662457331577\ 592165862654009353074536339073509549452634935890437501658830461806652168\ 103892789311454823125479004802883741419411344840101532941963936317601411\ 359814116395725827283497019497644281928966358815700485792928900644475599\ 011674796647678868177198081886517313876382523897383532862896083597206979\ 755042559717040921327367368648378806801634166592145939136471937441422419\ 953308349042199456785364143132458949344431612530571276751650155088240564\ 300118352867049335268436908289012837460469939393854107792506268151977146\ 297480426213476053405584752120361705297473888872235452044354136128627920\ 624673661113665406227912229337610956812780615978443800848631084387679522\ 323844084225171457432893452077097770745516372905683866326926542340817400\ 887599251216655142496292959594785215875769612373997552056155644636960799\ 530192967171455598772686859319385447082838198815987339764793242445316023\ 578222658270668008888914353671167477701887174162956233253504245261531427\ 925033712471778686703092057780798935218365898285212100888209418901867023\ 249931328216979614021239308600916563043358321638347235517034191246163768\ 035335224556668345250714797336172215949985964761912106464543814352830375\ 170549889743606981904782716747083858111039990007637144166127641028172217\ 263772829877377155666800207474988514049893773039805072903246981026100085\ 728194480085799414592787231354308079365556974469987932670204828930072233\ 213182077908654660667502855669579046884651232552119996190883596191352183\ 177752551788491751766747813233200047026958393726429168947648114918978387\ 547885273704275494595736883612115931183488448517502797627435360801827013\ 139738318565972350798612177397962834316831807962048871138377959106181105\ 352240012615295074258797660075066182413431266988647379378342137919613842\ 575031676857253773678597929867824367457155253173568623823218890593507005\ 559660731118685274576069073140564086455349563464144262275545196092926709\ 520844453471014057399321988447307305651333368762431046841925234157428463\ 235973926075780132006881792803985405351433293840258324195651335220414834\ 980304289568748455186494949524359760552578313040339690371520949324587120\ 744374075710240393928414886187629753000430404436291272509910588899245945\ 159157723632645438567431407051515992886859170392284487202967467125412192\ 006471505995171157966054770886706965079847924468689438297295289037693103\ 277989537217741288796571962889769481123360577141741417684399279843091596\ 590106951574987251498577498368645222775869222971925983322511241519624817\ 298769489484255267597666183130981884475750968736503408338119725678881474\ 525263738405216748074539531108418911843315032778333928009141647724614392\ 803817105647505409202466212271221210335689859382394498565976985789993087\ 648626473969885120017363560163274087722555410768174194886899820025655035\ 207256431700057569941877952516013862984097318391142761970972411706957604\ 673867894534175912353015146817477852099787284947276351874635861544589428\ 892305729129235676262871980067938346397792579677968489233656765217966813\ 142290195455359093242619692494016989383297994263372927271686975119992789\ 270901508869014453112306030060720853260754197202713386463304915498964547\ 027798495628510440361664308362634486729719061830300341463345382738291813\ 981861201073569542288353405071987761496892552470366999963283863533932374\ 757611776247055664540251334079422355931649821041417460726676024171309845\ 549548259564453941612280813315174134222963819064960184561708759201848502\ 092171405240367092787179083969740082838417933821880082823153804564093609\ 388698686847981614510293115318980706957750958296103695146227006625123173\ 737077906125826886383108775654396419646325058988410416128861570457606189\ 318294946056325024478261959145943595928283670027278014386680865388050300\ 358788269285404896513848585712467030000502108343978233592692911100633802\ 717991855934448783546448949873478010036830576445505897214444331817026138\ 381247222932838313774899600579852171881980018698299619316103480004972252\ 500478288359562482664626645701449374157580764884775447455014710653083767\ 283552818713584087030244981196494709414208873531544557358125487668854415\ 288736876603832015245507677448008776896669006269157049481399424659047480\ 532237938939769830615043785636713332511262424679076581309404206420097506\ 096377624823554468542243940078644555333609949585194165381874148368864906\ 365162091527141625126807738916221417089840159018855727544281500913789225\ 995943575817645774645030934198291379579446980428881722662786418828242527\ 779343006565054440810962187232543155312122150379351751433716105685926610\ 034053773578742624787258126268708971765187550444275143643000380467719278\ 982272559017348409261992852070761559504624639400844702749698824697598194\ 782533601420242608609973680021377456867364222600255466980823422133921622\ 681573461630862982130647041460160309436060608015632496200034635135950601\ 496328320996848625330904376748677979876145825751924772376463894981399900\ 460871760392479507585490901492729680426874340617989221767008901021711282\ 184215015393114956227964430541022372594469484836358504765957624836577462\ 230209605513991248701453409654358859371351678775468519933137224194257522\ 772509317553502815446657840319116283736098159477898419101391819095782150\ 132169181212358981013735938643218699079967379252578194335516187799167598\ 207119964088960647499378043720277273354826740062441038610990397510397712\ 307910196699168049682931485977978107799496887591490015938685646527408755\ 263793973196169022838465798585621380769356606300098101562249669701158952\ 364791780834270630463297278447583119160015777435703459343603731603868452\ 811180785138076208494955852988656045699911608938892888915355557177991289\ 338911023621000842459106037004144181801188123113588890812701667857865515\ 424107929952641785117387813848619481577871604109775832490108874165039514\ 490613901732531625126153376075665976834676394372484328769576804576856516\ 475813113288526660440563021138473675294552668888672753086826557786262204\ 045282123145353447194079470020069788591871714386826608302530542219344104\ 942424614924721675848904430680622953380153728955174769159162883388712389\ 255986873054307500109230633149113211323112946347133398520339639808757924\ 827577396289142459628611508752094150246885151761024922328214600510547058\ 164878274117648742299510560799452243142715628171993954902580208780780762\ 553446192339896717604125517736942645789118318705771709803751529191791501\ 043449178533167886367523544340619285216881759446627329493706169186535488\ -536 830685089100518743791078006269224374580586469287540489 10 The smallest empirical delta from, 100, to , 200, is 0.5204964977 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 5 5 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (55 + 10 30 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------------------- | B(n) | / 1/2 1/2 \n | 30 (3 + 2 2 ) | |- ---------------------------| | 1/2 1/2 | \ 5 (-5 + 30 ) (-6 + 30 )/ Hence | A1(n) | C | c - ----- | <= -------------------------------- | B1(n) | / 1/2 1/2 \n | 30 (3 + 2 2 ) | |- ---------------------------| | 1/2 1/2 | \ 5 (-5 + 30 ) (-6 + 30 )/ B(n) d(n) But , B1(n) = ---------, hence n 5 1/2 n B1(n), is of the order , (5 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 30 ln(55 + 10 30 ) + ln(- ---------------------------) 1/2 1/2 5 (-5 + 30 ) (-6 + 30 ) where delta equals, ----------------------------------------------------- - 1 1/2 ln(55 + 10 30 ) + 1 - ln(5) That in floating-point is, 0.510879256 It follows that an irrationality measure for c is 1/2 2 ln(11 + 2 30 ) -------------------- 1/2 ln(11 + 2 30 ) - 1 that equals, approximately 2.957409678 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 6, : Let c be the constant, | ------- dx, | 1 + x/6 / 0 that happens to be equal to, 6 ln(7) - 6 ln(2) - 6 ln(3), alias, 0.9249040789635498261 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(13 + 2 42 ) --------------------, that equals, 2.8862836267345560555 1/2 ln(13 + 2 42 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/6) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/6 ), it is readily seen that C E(n) <= -------------------------------- / 1/2 \n | 42 | |- ---------------------------| | 1/2 1/2 | \ 6 (-6 + 42 ) (-7 + 42 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 78 (2 n - 1) E(n - 1) 36 (n - 1) E(n - 2) E(n) = --------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -72 + 78 c and in Maple format E(n) = 78*(2*n-1)/n*E(n-1)-36*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -72+78*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 78 (2 n - 1) B(n - 1) 36 (n - 1) B(n - 2) B(n) = --------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 78 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 78*(2*n-1)/n*B(n-1)-36*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 78 and 78 (2 n - 1) A(n - 1) 36 (n - 1) A(n - 2) A(n) = --------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -72 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 78*(2*n-1)/n*A(n-1)-36*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -72 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2216611775882871983243142194613504530268835589475067121121860495556040193295\ 987232140353856635363349283004639808218445501001150293659846554855508036\ 591623564476803072520551039206549054347967064287748489420517262289301776\ 249423210639105045070453687598650653816648343529941755528133407259524776\ 893313457459270026626991255311856529697590491602621831302179746406799200\ / 7 / 239658557714094365575566417761455243302340395314988240560838125464\ / 765303311329757342294523856375256880209756349015201933833337252337107402\ 078621937351136252952080600683361179602281350817342385197540090410039614\ 391548634073933591237676867272045190733585451798826286695810401138710326\ 556752561747750947652326171949608468659369408566559910913382789434664553\ 29850836000 and its differene from c is 0.26894930225504738991008447140746847860166767556176366244522153046455172938\ 774163254362253207062921928653299219560932190895395917743460752249054630\ 378164398255082350287257905487230108384065035972650925715986250886066346\ 053388775993763122777920324439319600020721839554839121777381303774844035\ 960739734665231381962451364847856677611840162776605432423928735580237267\ 807431485744519963756492488486735246894422150333605274582719250936157648\ 171960753958396554872961514885439503533148747051061805438688438704705231\ 347412264835921570517338753924084122574863664479055206844748058789282757\ 007144346298027981037385430075284607791774647971947926138006890888271769\ 211766982107937100116858593402518547021566563631005816641446963142743933\ 300095718826554923592671516451372641477710704560526938305996399820178763\ 887091495274454726282137656229001660012287793404881888744987835860353358\ 745803559184551897248352597901089144852069402297950071502345313209683127\ 335530661132809951267175016128516849937830608852176966497372867588431477\ 966787334637528552552180661429932572891977903821721712523320055778759839\ 996107075980302034326830232844871116851966087625588596382358498899148037\ 152607381403572265491251262773316764215076771561094080968692797211213885\ 125177046403411933954406650589084746138787540936732684860617763672485016\ 016996332621841256206244000331947794012794342435423723634023064899670133\ 653929960002853959724296161093933467651110817603085348801187134092601089\ 110248581688934294863194730023946689689451825615000402592869806799051331\ 689541780084435010363875648650452450523346553912456861927697439266138670\ 225969459609873155969819949936175468966818170903209928786144816532141103\ 816705524456130178032181432108099504480541110469547352026781835968436631\ 957043484064440222297712599771821400181979464190767942260640911215825910\ 671885757676668058689141828233557181816333020439423558960503399267562301\ 227581027891504879117248339637127550692887137326301954782840585253999637\ 497574334657400373315365566084316091912277558158520771205331821768570284\ 725561095830520817103911775866185828567355062830166500619366217325854483\ 623690787246339518899156125237896270958535247660022443503504098984817715\ 821530347824087969386867960185512627816457200032753339410364555273489260\ 447277886142534961368607623109220521269093080963088339744393692490415759\ 706942792265883985766194038563788151424127451039346485469219356871003340\ 017741473312939828826491349121927246486910527551208035296689207755834950\ 413719728909150566533572761915773019238201140168697412178018311393433732\ 672241443544713820336046618159687678720711372233783665036117236041109627\ 845439243886753378006319806800241602443137413682503562654486118190930590\ 397529782402428936560818495806302201239103609337603019681211820281238471\ 790305018538709266461673686477288551184074089696998101638102440880759968\ 999002898584516425249980608260633968600245198359325669522437641618752909\ 208875768181602726391367698519586601494709392650082535188875805008131874\ 961596384092234685014542444220955935275696531769229910638104798237982173\ 061280968485582824093252836518330244696797935011369360625005795397368706\ 669989991286310852580461974748686096402005176058429113350707144684967967\ 612220286376605450700609220694368400247287782288711152511760425633894139\ 032816808835873332907639000653250366210691123396915280757888244772383500\ 394518542953097349672038406127350586011831958576721762055268709880383794\ 624680496972177433354539196642235738061849187281147253952627198408617955\ 893340356207388145356858335837198981532774477752883191971619455014492112\ 585272576095414017479206128915323162630542292296953698765172703971648197\ 910070719059568880975831989326627289267826257695332664950367324067938818\ 517369863354030393668813197030940772555222304046747343405962772898574160\ 840482374020574065340322454797945461743341081379842944496834614674498558\ 707032100352042885423429451672506865122390920534502971410110097071338456\ 143130239142559386959006878506447848889011031230890860066470530148653889\ 007892713641964304443559695672933326095620888306527519260402562734246880\ 469370806941557024920684063670514098647818927809886673632688765979932144\ 483942107552321616969777665639376123239409286245597296171849382308459940\ 493253113509436977539998848857606922229066272885272069318337182447806675\ 238086257209625326151434823048339842035758801824112516293993934361298919\ 518420923441597549699392168512644400544120399020769353342388431922360587\ 073955180114429370489723007957266862622896764072011648658328632655626279\ 970091789736299834595794464006703319739043177782926694533119061703863934\ 269765323787156201841495304778072624060526915279826604847705328406975494\ 147580265925574251455372524581573050844985602558559830120267636576323905\ 414952349557092970547679458897935461436535608038624935469236402853188136\ 344233068110986457928476081355195737929499537757742063002777615170480813\ 304899849493562417779583828121455097167457502462696511437965631051198036\ 700677929654627055825126886775699937072562950573146180915449841406369893\ 867177205492225632385518943979379080370259142150412976357689804448364056\ 912419502102189400502757014346600264454859292961334616818440939932857877\ 080982828978107829202892649893350029532830238133853908344510172224482599\ 638307684803161477404419597754081390922407939535587625767086138155670678\ 844609335574506122774151677869241293671936935329363488620689002690757496\ 841803762629328559465880977974483052666762186265176163390869853982076368\ 695345008947899729644077137415381387206236078799566964292285302802365778\ 817493695479038687532563771244480130691353689293541916214542499671135796\ 249302391697340764906845369089103091178905066331044765638917683226628037\ 096565387732098227665074656665750746890265320979392068375018526855194276\ 132276599241067007796494396939184155510387749159876553262057086128444990\ 937857915959949845441550376128249874467095967728914344886021813042671154\ 023625322807900864036230092003796200908096638339210659589619425180880693\ 764242892006020055699831094591798049849603531423059983725283799599575658\ 902088994566481453380857698182408485717409574640902758727801004141093126\ 599270535068144675085877803610915860552053166846860242101211110378252467\ 010493409094504507879556709252632985218725407245285533765022064348224468\ 852791183432899705280994565846905874554256914834581474028321392824511507\ 434403764235268548437606755785065542301353620880365527664804856749718252\ 680656934724872644576433505880165924992249246709982821318263102929184745\ -565 4008465938813236665370875 10 The smallest empirical delta from, 100, to , 200, is 0.5382153614 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 6 6 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (78 + 12 42 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------------------- | B(n) | / 1/2 1/2 \n | 42 (3 + 2 2 ) | |- ---------------------------| | 1/2 1/2 | \ 6 (-6 + 42 ) (-7 + 42 )/ Hence | A1(n) | C | c - ----- | <= -------------------------------- | B1(n) | / 1/2 1/2 \n | 42 (3 + 2 2 ) | |- ---------------------------| | 1/2 1/2 | \ 6 (-6 + 42 ) (-7 + 42 )/ B(n) d(n) But , B1(n) = ---------, hence n 6 1/2 n B1(n), is of the order , (6 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 42 ln(78 + 12 42 ) + ln(- ---------------------------) 1/2 1/2 6 (-6 + 42 ) (-7 + 42 ) where delta equals, ----------------------------------------------------- - 1 1/2 ln(78 + 12 42 ) + 1 - ln(6) That in floating-point is, 0.530142968 It follows that an irrationality measure for c is 1/2 2 ln(13 + 2 42 ) -------------------- 1/2 ln(13 + 2 42 ) - 1 that equals, approximately 2.886283626 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 7, : Let c be the constant, | ------- dx, | 1 + x/7 / 0 that happens to be equal to, 21 ln(2) - 7 ln(7), alias, 0.934719748371658362 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(15 + 4 14 ) --------------------, that equals, 2.8333040239081998161 1/2 ln(15 + 4 14 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/7) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/7 ), it is readily seen that C E(n) <= ------------------- / 1/2 \n | 14 | |----------------| | 1/2| \-392 + 105 14 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 105 (2 n - 1) E(n - 1) 49 (n - 1) E(n - 2) E(n) = ---------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -98 + 105 c and in Maple format E(n) = 105*(2*n-1)/n*E(n-1)-49*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -98+105*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 105 (2 n - 1) B(n - 1) 49 (n - 1) B(n - 2) B(n) = ---------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 105 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 105*(2*n-1)/n*B(n-1)-49*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 105 and 105 (2 n - 1) A(n - 1) 49 (n - 1) A(n - 2) A(n) = ---------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -98 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 105*(2*n-1)/n*A(n-1)-49*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -98 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1603744507434532924363226312146417401792909312906571619377158782941706025692\ 536600158513591862073515590718421130891039894158933249457820769996827654\ 116240615180290751830010578350260348782333093338455974405543974262100891\ 470705892636853538164804834225797031146128707551527919136536364735407483\ 520685611859798936109128635036986205765127348468611651376390767979619293\ / 1673186831 / 171574903625215848965330410366737051954882324815239080152\ / 005791043319168136958231736834817367473194047697255872664915582633609356\ 105124228955722924768195245607886747928606169203749240203593040070114500\ 928279082829842829424566687079227719576660273049114411034752793769047404\ 511199071404465520004349572468819305514620803802122923119067499084370640\ 96344678565162115461192071360 and its differene from c is 0.32429306864157542519359921183809465047548389476082709383040398072915941901\ 858707208230805199702177684975231240701554929525861988649757539525250966\ 757413189862782649742009305592099533180143429753226293942345370476207736\ 154349467882370708684814779244001669609816335944634882698264367864881301\ 064460507115934692896483534473106701954081507338904005346427175064995650\ 751654398085212647750245794311302123491509638994831751139984065212672657\ 114903159937579895381756501204168829916147639902643563345566370409421217\ 245933851452669846294335856771240528688405858744696317902394747667381031\ 429123570997314588020449296437556234834754471990879071686512173898193610\ 039665243595954564507143983872422248194173823564795357549879124016941307\ 623974170769595573449734405703810089751248788739336351779488188944205578\ 587615833435684275820427398226565141510490902220325252531828429796162597\ 533940637462021305696676308809615603009414302402720360561897981247300433\ 951847128948573512932015806716039072871912274791341885931305788393400628\ 046778645330782729373782875565476661767680250056835610610494330455890808\ 952245774255261927952240194420116377755233905037882160899024737929304584\ 479078316546669116392065939864891968938568003648436989391154508279609595\ 662472429224333554336710493256612034803778779696491769653690233781181044\ 272569953176447170922787413493184026278440596458263735104302678623651505\ 303330011019313072962951918396970637631743138489262234174495635555597008\ 215816027580740826036180437769877651745349832341271879838826260782278555\ 230799675070402580435349608470534679056989275240626082387777482018264971\ 165021629317946658046846996741402984439020168414848035116980639990408915\ 970472714502781985781917002998498789395496609157364801605297501000515037\ 484308341720864332900870186586655446231504593649749485732285898861217057\ 463016494765312900760746579315338658532606199832768211989617460158911529\ 178157923856666260783251176776709764261175201888351369367108923040828023\ 992659466738910949998887060416346145935259191552617577178303678723954909\ 358481965611053189897314139401309698127687957046766358978459475989727761\ 527352166629334732514147459956088992161768818911111665252289035645612692\ 951026538894389128139427763625357690359658651444512307507501325265435402\ 889816945677024607821276328832327459522331451592492923944150400448520062\ 860639357862282610198461733962494060591038909237750964703937927034326899\ 952689258113219313409700405152358309574982035540991018516830842612746121\ 415755798473706693004393228920917946247222874594184818051009180097033732\ 932479849559485249536576768296083985762638967472536351113615402186171899\ 511926032059327677661992893768742145455818119033864277065553119975567711\ 345301158084317843287999744241859490437489465359904023953082216356591382\ 524726705362337828817217263352124301492216010250738580521736603502507745\ 820509550060122139545650921413936878643413699173717699429278080672783685\ 153000595732269652980900711336738808651338477798841161202359065073386160\ 090233882108749311640926551163971793942974723564187755421980343490625195\ 899876415464977359822939235164881312258735657208502660689749773544322010\ 253595283609064031653736561365395623505572313248417984212993024933018715\ 300252147062746048364448225269447049771376040502642074490636887160329853\ 211506517679242099930991311554946194661180672549599944476298744497846396\ 852352074464667065326829814797535717494494079990166011789641008725928837\ 474534945423541926994812276804777018522087329211997037312534371885297869\ 749262097964716887599566717580665010144767438300420911388059110302638764\ 393139345074182549905243668070619496170475139982197435112017476262210834\ 415968422633095029399303927847172430719892024501325527256475204567487491\ 961634594871104281358352999093326405557352934686198418707604099291580623\ 474779941163424588306764757145405882762366091126367992234259549881439027\ 119830679718942259598366100013042966981191858797465179808307800709289101\ 052295326933724096825652086943876325251991221028022524811339407291342063\ 694854855592801315646879048676112497943547707000446884952882406131364596\ 116221459986235293102280521998567262359188049647513909275956852575084445\ 812547550545658692897795016083092397076596904757953182569193191734427647\ 947373079877042133690451637293801672465155129450795907443138325811548135\ 461785364797656533780636874856569630303106723242543307985877653943941505\ 073958708755076543421772280074146318083866998988568874101819587940819749\ 357344813919863256615467060847822364350067371843242026682617659161705373\ 914360325957382232312542173097187196863788816531328680454046846198528645\ 373515507582066270935249779456313520559040493031748064500712792413409716\ 143836289425994224834044650476104078743221014758734100993059888553076777\ 758222836357577386619950106177697695043911113741547680012073878374256506\ 040848293926632723227899100385594026672890612018857270925268822870527019\ 433667419585101348732845525370096638403721695225434961484115063862980790\ 979213927383337257929046609718298370756384701217755164188222835284180697\ 421440157236628198881264881716494845124253834053729061882752114222853733\ 566356194617745928156253076351318159343761544092985337726213829612819977\ 546328536478783248878500519401037825046968447358430392203758035674857005\ 625871545169729852157473423227244229095503758081823817223678728175090559\ 048667654526397893857976382518109601176806161566685479648878696174165503\ 651744815242820001702924830680924616634669321756195726756566561808578738\ 670065901376008479891363033536216010661893227668199624955321826354311354\ 409889430316553117820630848813167146674042579115411035863912123468127522\ 757652980781199054280528763812944072293606457286732777957906887931950736\ 227097499382517967234840662321344344807714579673795215611181705399324355\ 779864977863872017077190508517307275482468493809897678180170009074644225\ 227753175861291039231065410669203039925856963368553141020451273155874354\ 258153162265415998684253564474995928578829981241174480591789963786247018\ 545922941278320097278547751485129650158255921267688300317770672548596410\ 060636783740337921222918921733132412467808091540509389923033080573255238\ 275947153797661715835493037976728132816077255434379052567888246033417918\ 040583009635600376732579386414565656092933703818690685340516792903417247\ 979656182427578497660022486103871593902850934665223910095240297346772865\ 512267314051454068951736375787305743728556444301012217119909170253140247\ 310554680532207001860296728877232763647564620842155937699571996799647702 -590 10 The smallest empirical delta from, 100, to , 200, is 0.5570754845 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 7 7 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (105 + 28 14 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |14 (3 + 2 2 )| |------------------| | 1/2 | \ -392 + 105 14 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |14 (3 + 2 2 )| |------------------| | 1/2 | \ -392 + 105 14 / B(n) d(n) But , B1(n) = ---------, hence n 7 1/2 n B1(n), is of the order , (7 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 14 ln(105 + 28 14 ) + ln(----------------) 1/2 -392 + 105 14 where delta equals, ----------------------------------------- - 1 1/2 ln(105 + 28 14 ) + 1 - ln(7) That in floating-point is, 0.545463269 It follows that an irrationality measure for c is 1/2 2 ln(15 + 4 14 ) -------------------- 1/2 ln(15 + 4 14 ) - 1 that equals, approximately 2.833304013 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 8, : Let c be the constant, | ------- dx, | 1 + x/8 / 0 that happens to be equal to, 16 ln(3) - 24 ln(2), alias, 0.942264285251067636 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(17 + 12 2 ) --------------------, that equals, 2.7919241638857524935 1/2 ln(17 + 12 2 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/8) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/8 ), it is readily seen that C E(n) <= ------------------ / 1/2 \n | 2 | |---------------| | 1/2| \-192 + 136 2 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 136 (2 n - 1) E(n - 1) 64 (n - 1) E(n - 2) E(n) = ---------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -128 + 136 c and in Maple format E(n) = 136*(2*n-1)/n*E(n-1)-64*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -128+136*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 136 (2 n - 1) B(n - 1) 64 (n - 1) B(n - 2) B(n) = ---------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 136 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 136*(2*n-1)/n*B(n-1)-64*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 136 and 136 (2 n - 1) A(n - 1) 64 (n - 1) A(n - 2) A(n) = ---------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -128 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 136*(2*n-1)/n*A(n-1)-64*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -128 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 4702122494681654856297705303820783855062698579361349394590623537282811207936\ 572183122651715389302077483531305335608031026213605691682005909865018713\ 732774617233500197172014653763312998594677411350080589328398078633041423\ 965266019779644792509368336595209699235459085767421381341087788857405807\ 947221921342953649086165987071805045286552276803402574384072912773621193\ / 6985090120372449708331653 / 499023741882434580753105841106916971422205\ / 640133599916846376142551233450790769558866238288910381651215315041390397\ 868857550805427812610098344905194626133351855914040032696232472030689058\ 617874784810073815039487408825536466931253616830580616931573727644805629\ 092230534090555014927057848149470682143586407868239495906276423820492320\ 47484821272614382241068661920937390416059537099771335117000 and its differene from c is 0.53521480750853063867021683351359111205744020087698157540036979511291325064\ 716572205501596999141980747034905843898832734863168784631887413651846262\ 490143120031395288842157219446825971635090894417951796195509644454754718\ 286836746386155890635061736184529443374625357665955692853559711180345138\ 371031045284520779002412075000695688775088967377071781074467777504036929\ 122424117739378006016505512054270221365261850084437827854071929977235688\ 013488287150319830407886227041169208581667027673045916899035185758392486\ 315389492473656519772086145920183264238833235776611824990227535836723616\ 740638199117624760785198360069598081770226075810285666803228977807927408\ 246831252975579078983369032196879206316242611426637944345224989153986907\ 946016567143622157966053809779077822717383968044959818203725075131568673\ 559551998756447812739676927247080407731045967248224143799284275677445292\ 537915412667702359162858741564701485474306057721099927628577810693962261\ 186653509264281067073994228532307099730380196242956807572417869876442162\ 794385473620230368225328466871358710398260032226706677846741740832779663\ 723057226447204602709006152694172972424962280037566667636059019907866973\ 016523723853775217886237768184425179517456910944926775636079308572563766\ 790413784093583065154939925922914996804033350180518803272163423089702262\ 990901682649535574904960837629223239299885345635341595448919908572890302\ 167657511139134393731706055679977924529599709282916246199436673262357980\ 176911469173210162343457515378950089495609774746976970627786664848420796\ 070188971994400455742681661013632308057902228028970981436289368548099594\ 318677133024970698572008548026615535313861088688674110165593395608703784\ 689980503633024383283496948920803272324156004824636016665282447249218150\ 701888825745274838817944318763643398198162996789991475320547592875413143\ 608396281041560603788207412298805808349345301505347488448276333498508862\ 566515835431767778007509298508448317884765695919295764566006283494631735\ 550768639224677541888120890526816544835666143836273189359696222627898289\ 727030662793356537393888520059396582787192749966533238122892057416700468\ 968184594251304702332680240531701057894444574785713484928084693255279255\ 484122720552072635283249036737906525018188011122459197490195478991649956\ 432126300539323959591371254202935950534000261005860028328509618791581831\ 121725430450209883933718769261967727634177301578435630844451100166044091\ 541453231746825118915897918043527199471733097025340261566945621074892679\ 149639286258442800559305153520878026975828705703813815564852909184688826\ 573576364716542277378138822327751519857425051707537446816867335196393096\ 769243005015351645193513908983786347755590196379417422000183873723593068\ 186924665222881431812567376416042551119771756131893579052059317248837200\ 529256543992586705276762320258124995776956480594473417487698132079585458\ 852639225762639976359300889616201737433614642474232307036320476768112945\ 212593499326314998857579339930403333905303133037293103656178632611700271\ 242252554621539433858004547324829371244590014132778001997271251219317482\ 919523592841771587884246228851062537807883467201914798686630596851083911\ 099318118934098485259486114407391723262821553817042484304362323970960216\ 841730423030525972838694822333331800763936286668339795720303410007856874\ 919017028747178198218030118572169052727895314476382753659363343812124447\ 892219850368874208120038861553437886010152196516444714457864507631265781\ 608634681298923955013390679276457380126646256883715806654427084699346381\ 960315582337591584647721223612009141489301793423774561034615811784492808\ 367146441207168337156174337130164352256011714513794597882631718019610643\ 180119093480229419438538422277605335218118183902787532220045579537079738\ 387097439074132403672447780318894063580599660511908247397559442090871403\ 564677711310846003493884904347703708465626291074647486072148935338072062\ 314971245313968104529885767672373171290638100596827899808495396549997239\ 387952467885459396959027660122003181337936165373295242940557680266055660\ 997156466757084013424930057724487598078354061501533645221008422042494363\ 770000958146878012225297289942067557793947535721015673403714998733422113\ 799737579547510312822680280156251330709685613574186062376086096440147196\ 187964140888194553783396530097865867391041552259874293201809951123319764\ 228654018891312203338647023849327053168868660920018341846741408657599559\ 736086799656895863049312690856097838389632400987762233286913928819706530\ 336775903540411440323833169360302427707880743613594218081959619612639560\ 738766043782161270230077255838645125359206216320156857333738334934905279\ 055861223140730829509692803437543233701085000967421713675348101640632768\ 449357237752222966183298397708136204042921062199100002441136343359753381\ 908585931005278824375640891135951255707701108204515623041702927681849603\ 759177089617236128364033177521755338965674172762262851282084580614136446\ 565723402182906843266711283780220109559700959065349424736653707352420770\ 327492037472335783756025204418462102510235014775412379178347388470474714\ 131062134688800994339089574814245996572525486159989898398351574087331735\ 082482296349016414019837826679615694164676256483303874481246741610092806\ 164765515014503013339917252991138503229002561614236645787784537037134091\ 771309461370593410112060591517299381181368646755284730655424834964450739\ 035018730314232812170893596431186868382556671619027488012562470409935799\ 620949691381517728248598615446904446596026574078335977996682203342902438\ 826949731922054540609559865443533652521037477697805521251759678906688029\ 701305066158951268900157554973821278452269064546654169967199703818356886\ 185573260134633915019886241295414208692656697747029651434049524173983086\ 571403792618284912129632941818341080801605392317669818814257725273655694\ 726765691939432209785559679519726379504507106172494757332279466849897374\ 812018681443863087807114444872784498829834671143142911171392403984730263\ 082261969070946455682877661105667369459956469202657237452399401387421186\ 846570800836514079269463818949819186894877951005379146737033811671161894\ 537362444687294000274788035272964242047952311382589991867010712813634752\ 912254622716084454090591993622720778930290504183223044376257616688979997\ 290802013408390451505498806327231921360602614619982555852164770959185050\ 088661974833713786220974255607773826158491249577572470463854710928816849\ 169635389596736246231063354444633711978262760110433941368911606513117540\ -612 35288670790751855661000435602010786118762839685186 10 The smallest empirical delta from, 100, to , 200, is 0.5687325130 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 8 8 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (136 + 96 2 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------- | B(n) | / 1/2 1/2 \n |2 (3 + 2 2 )| |-----------------| | 1/2 | \ -192 + 136 2 / Hence | A1(n) | C | c - ----- | <= -------------------- | B1(n) | / 1/2 1/2 \n |2 (3 + 2 2 )| |-----------------| | 1/2 | \ -192 + 136 2 / B(n) d(n) But , B1(n) = ---------, hence n 8 1/2 n B1(n), is of the order , (8 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 2 ln(136 + 96 2 ) + ln(---------------) 1/2 -192 + 136 2 where delta equals, --------------------------------------- - 1 1/2 ln(136 + 96 2 ) + 1 - 3 ln(2) That in floating-point is, 0.558059386 It follows that an irrationality measure for c is 1/2 2 ln(17 + 12 2 ) -------------------- 1/2 ln(17 + 12 2 ) - 1 that equals, approximately 2.791923987 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 9, : Let c be the constant, | ------- dx, | 1 + x/9 / 0 that happens to be equal to, 9 ln(2) + 9 ln(5) - 18 ln(3), alias, 0.948244640920436711 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(19 + 6 10 ) --------------------, that equals, 2.7584684178851320662 1/2 ln(19 + 6 10 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ---------------- dx | (n + 1) / (1 + x/9) 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, 1 + y/9 ), it is readily seen that C E(n) <= ------------------- / 1/2 \n | 10 | |----------------| | 1/2| \-540 + 171 10 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 171 (2 n - 1) E(n - 1) 81 (n - 1) E(n - 2) E(n) = ---------------------- - ------------------- n n Subject to the initial conditions E(0) = c, E(1) = -162 + 171 c and in Maple format E(n) = 171*(2*n-1)/n*E(n-1)-81*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -162+171*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 171 (2 n - 1) B(n - 1) 81 (n - 1) B(n - 2) B(n) = ---------------------- - ------------------- n n subject to the intial conditions B(0) = 1, B(1) = 171 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 171*(2*n-1)/n*B(n-1)-81*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 171 and 171 (2 n - 1) A(n - 1) 81 (n - 1) A(n - 2) A(n) = ---------------------- - ------------------- n n subject to the intial conditions A(0) = 0, A(1) = -162 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 171*(2*n-1)/n*A(n-1)-81*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -162 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1920904386933034853283295183201076645126780210520249896691456852738734125289\ 150112424479556360451773147969270750581537030096327837976384331242690652\ 722323299557834631476514159899738647536436399323085378032182172731465729\ 456303997540970214030519048169371759867060572653838823467912601841283754\ 206312929986497874258848507466759653148232509235930406274590881554055455\ / 566263911090962012527493073950815487 / 2025747685817092850572514699123\ / 922600498158395898564339859361760999077825240789485764548998051543761794\ 696452650199607725737630847426688293840376077840650690514534005616119239\ 638138053939172387351105505363584478760907642210707316107871595694847511\ 557761297366894436692605995631998992895600622691551281712996229113580047\ 347807217054140687486094680470315114783897897937861127981820014212290279\ 011592000 and its differene from c is 0.23954509464097807122985802811420407013522088823791812026602293639538294115\ 224775339936702646674404857400893447342921524703977180979773446418602203\ 199912562409110219989986064266445936328653159185008473681289452870948847\ 916161682900729521490313702667444855789428637527246348721572344247543861\ 421424236132860838141794935670689368181912761105323597104911224060077066\ 357634759989728222059564733649447644596732464229471366342029037386975301\ 153881595521556091927263282505041433562516530555358315031918229166607911\ 423968099016924812719137229039311568861119303547219393534661365304805289\ 366687057411463589004362986125294184039292482579569442398515960343719322\ 408904768342429668102834105655984904848645093222375407369097166523229243\ 896404151788166187751971688125287856906321521712085477259550012947690583\ 163247941198700135658411874787130515160576613947171964264094828800319544\ 492309075450022070363043847013771926366805692589565945897860035972209806\ 912527827914485260669298945494649346303287740822733573821960012383531597\ 899799199731582779276943448903794198329293950175350338714960311377083833\ 294861352973303624620518055726276787123826749125751958430109481677944977\ 508103984181765738183212297283880076043610698261405264729307288378584633\ 630211287540531604237398383982762167855814591284083910635867993361332427\ 908800838071854057171503946264860718571247598466266655502821303604454141\ 586034365078185952138874641196308644173424769541127040263026117122092549\ 893342178811853808485819004194146060687037995822527472735548086345694845\ 468955901479582730289732695887776954687256607259122682197378766354681524\ 938428974954518138680597833199671619206105837087296341141055991012818560\ 720434773033680532506870410346175747296948553052741651946499054130698773\ 695142190633452998518597987128168775573153614326101943934912432348331765\ 854353249178963113441917495941685493952935090896948195982949130305436560\ 031478304555406159124182401770805752891968190257943359126662984896040971\ 420891370971012556378815912981235711795662111557423861184668575554920659\ 304834740820051038035753930531510766602889123529662315969455627040022497\ 209169875631228243068918065227399241273802152864023534488499818012227260\ 644921660063754031205860497642713805150529451791734927807880497491763700\ 707132922287816722233600376623776146277612936694317566692629445267768402\ 206733707263191914722778427544130107844339471044380284537272629768170036\ 800568741629590790175966422797523574429706253088765975439966618047032822\ 730264513249495682162477660674827320797740330005230126981755241632770931\ 286370736697914457088174179690402102984295716984158017507266689407408265\ 908227639694534439523960008257818840670973611599676253809010961310405590\ 658469370257953190744203608158220528632286371156013338652248160475804906\ 329596889282577486017839051526066069143062071569515620753128408419050776\ 028271539169578608657927732695303055915244433039784048909271351684192486\ 595500977245150122583966079014347045247178672147364903434867421784097483\ 488604041451422318518440184582990580883659327572418635446092630514281926\ 288588511410113015028217215261633371460301705919679455893341824945433701\ 130516273503792181330699391904660511357751556284681392587186675615282419\ 445454493516383191374813882368086474979432902824116609808052843537348522\ 862357536416373756525172595456350154741546720377151599946320688717329017\ 795307201988644186711110787459003112888893625475060215694505063116200459\ 420913691142557277643221764572436582292799630750816128057490328390578036\ 162506993492703329797770369912754478104064800096054083996926452125416870\ 808638453563332803209985126934915041747753090590552071653756659200322310\ 597110413617228725388843844989954593764834911612758023770070281873776765\ 347138786359533869580715442373481579303105327462557971803445483301505636\ 859086232827294805292572848469851780050648375461917472965894272830939564\ 871325815867782988254247312670969808903547569492816211347396400209759643\ 985389881855754366367388907161722821128199786292528509449220578195617805\ 708904928659618343412787110415387107850967364083150521470700429808771617\ 300099108185134369960010700016119556540247496780696851004155900245603443\ 833140064951056012109957289356474629627180730395297316130553251014783453\ 099831397192796600134968233083586859709102554239696951855868814501320589\ 479162521623540887983554396524549150080103665176124649778032740081010765\ 732117563337494474474871814354588296970907719479139853374657515944031519\ 998195044323424356220439301212526719705413451354508934826339650122148882\ 065383456634980690951047855998925312726430025512462738101250655632548619\ 733497858221954064417957024180435864641464461505025104786091156773634359\ 583826955928771083966938050346970450050705219991739151168296968787383465\ 538745412293572443735299759856283504541202879828328979502113949115447197\ 119559561136639238935111827770229773700376153279748685341085451729919801\ 665451043286167107422319853131696909197464422265800354453415767047263696\ 225169473358289730479345376350965531419119925697920397652933292310602804\ 144646237337062916305008137944374385179881882346726723008848247797492565\ 636144769764143017300415445258832077378665046427253822772372807297036083\ 796922873056068218296781922342560282720023757211777877551212707144865288\ 747094526943905659176200717657608741919691307509345123237231436343881900\ 114020522965520716206514763552569520063988840632213532046365333128918022\ 978516060856707212625394188593465262106196423218359309945624105697353415\ 103063562794871157019778600076292305203830645712047295544650765921560615\ 180660036268198719570831830089481314921814508507531394338253855471454837\ 596632698097971508641055245044786180458045770527020471558157828777021515\ 837911941743931370809855609217898817900814800127575040283042135836445338\ 842997301996838879149813111587816565924815020479792326661467027909015391\ 142443398422271412006689969812702977101359586590345385099560449504795021\ 511747479272286028781913403279643801023443765621236941244831873233374180\ 684486213555355457579517428027076547096448194117600611948201043350449706\ 563912808167990795019734130709782842940228672233166605262783813531287276\ 009777308346642970285917686693522114918042244676254031640360873757495862\ 825451301206905280339534902298924581686994157728803848809112884098086754\ 828696631836933344086927107739889797460147836570137233715101729980791187\ 651342055493822091612717088294620361858638420275888973410344830310293413\ -631 5919235832989290873884720632000 10 The smallest empirical delta from, 100, to , 200, is 0.5808286542 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 9 9 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (171 + 54 10 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |10 (3 + 2 2 )| |------------------| | 1/2 | \ -540 + 171 10 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |10 (3 + 2 2 )| |------------------| | 1/2 | \ -540 + 171 10 / B(n) d(n) But , B1(n) = ---------, hence n 9 1/2 n B1(n), is of the order , (9 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 10 ln(171 + 54 10 ) + ln(----------------) 1/2 -540 + 171 10 where delta equals, ----------------------------------------- - 1 1/2 ln(171 + 54 10 ) + 1 - 2 ln(3) That in floating-point is, 0.568676687 It follows that an irrationality measure for c is 1/2 2 ln(19 + 6 10 ) -------------------- 1/2 ln(19 + 6 10 ) - 1 that equals, approximately 2.758468428 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 10, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 10 that happens to be equal to, 10 ln(11) - 10 ln(2) - 10 ln(5), alias, 0.953101798043248601 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(21 + 2 110 ) ---------------------, that equals, 2.7306997777303513493 1/2 ln(21 + 2 110 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 10 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 10 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 110 | |- --------------------------------| | 1/2 1/2 | \ 10 (-10 + 110 ) (-11 + 110 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 210 (2 n - 1) E(n - 1) 100 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -200 + 210 c and in Maple format E(n) = 210*(2*n-1)/n*E(n-1)-100*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -200+210*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 210 (2 n - 1) B(n - 1) 100 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 210 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 210*(2*n-1)/n*B(n-1)-100*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 210 and 210 (2 n - 1) A(n - 1) 100 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -200 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 210*(2*n-1)/n*A(n-1)-100*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -200 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 6438354934656957926038904882075937453882773076441925971663584603407139687198\ 546025661546337801285539702854470107006597417186360423303983262381884751\ 531448282914035203543461893634533733125426062947729601948644628740848476\ 349416267212399985078712042259235081524806307245309437092253964501192526\ 660619593483427537083917968273975492057354818359322794392333253490725037\ / 18597262563690065574450352025069182694516539 / 67551597823811960951788\ / 880470343395228971878125163909873474734715862138304556470272091404969055\ 248038994511790909573860606880896461929233699332040037204028891875681837\ 139577435652303345541641806054784449670465331034423249247573362772023442\ 076295267695211269071555711899713011271642662876541009202125740821053253\ 198948126355139830721553557291795592012822449581011823907091959136243855\ 5553935358758210684098400 and its differene from c is 0.94076687968538513608574144819488973443899522327114957961635148090353301123\ 227654747341348070713018875666060374197565603116793154157778595892524849\ 252098763991014388090819217573285334227577148059458138748141443139126001\ 297224463850210796802459304044985235554045278406152212034865506533366062\ 779443928839697600197579717933149519694126166678402458279645657601041813\ 735291182999204468170917714917557485297729606506927428995515571542021942\ 848101841381426111760467415942398269100829753998196761926513707491537452\ 273770501739909219382453640514002674847673033216384888142963759974948181\ 097288808143069338303775319568443179462247190285838289608711785885161324\ 450089351182818695468998432224586948914840911341700072359690754308805735\ 419266402592184495905069302322101595853797385632703182067622712625827268\ 785245765296893381838891639854140015493262705508662212449699444662457177\ 967570403686862948829731230120930941222730990270111641301801056298342189\ 889507209391271229453838935729461722973937224980665376052002704150765407\ 995646509064466677568953742712661923342134479751772106827973357706387151\ 649380622896131658428126322652477902412645091362563673944842198419178442\ 350812313144035442961151681760480891831989318195412506953582916438150251\ 081955229686214873313967144152516835646538622933609680129688272281520464\ 534811475162664625571217167044931600753313777287791912843987419259565454\ 749015200960687799874630851024221586230010897790074449238005474615895023\ 727866193130856589130527894221335558140410946372935605606085867803992640\ 891307221801180466818227310519968399766558010785230377140063339866232843\ 779511050022045965689998678623897606657720341036976200960133315626142453\ 334582947082897885611784924460356573162507436407272853190313188340603707\ 159485259881696691258524556236656774058487303822259129656503052772443567\ 908417629522046231328686295677121351850149253169844341400472997317679138\ 124163041229166930189748109311820484242044537064926001629529805675822871\ 244697924981738718598950192001127859965089082618920631608751046532879140\ 502885256831656386685298968535427386137539438473779262495309874492850248\ 192810841692757582020505020930278011014323058872555966135303405740182028\ 668922336492357627217774624981838200197571107834029636770870937101244837\ 205893232559986239742134518340393131145733181796613958209387975810182425\ 893322011417401425257007225749148521967175089758207771247763122100856533\ 029721153532818211508319077838354229960930001832086468458107721066492145\ 814915529500990859056175842867014417355017218932400061273013711727081589\ 546258136914499015836968405109153488074440117551097185463797541547912982\ 587139199954779622208116806633403729076014225933733078185954565755875865\ 525331408793709427927965558246477768974240895595120841452246307362240032\ 348546391269275549358760557334699816336784925229046239967664419352567937\ 646124278288650894424274914136065306745507927166589436490759619545662472\ 210498533392929652750956197582022829635214186291806642052374788419785840\ 563963416184864280056665061056462986388236205025820866686898121749007502\ 887897952016554314278987740835304059671918469090057178026920667547002817\ 312873560318238313635987142410930317982534437897308060512162839569966686\ 215034000160548745346628387822225294803126375746327265924264614617388769\ 122979872320487083588674051375667978425532787665176515161233579270171518\ 844958270813612489785351242823126879914755082876254177038226584192983561\ 281163466505804518157670138021917867508917344566990971695196886835095000\ 745021881383860109751699586200464818003168877519051101378089648090687031\ 958824376539985021093651326609720647430563725109999520751453264190530322\ 934226931790611891253740838759556104398650956595759133716818140484550208\ 354772763934537141746302999460369329629946880324615049367627759954923220\ 771797581029913695305136858238935551017810824656847584216750731350772841\ 298890051076021235126430920660345725143958696926063545754649111579042187\ 317978415864706817146724370778901275814693085735991711995410235855218792\ 130861530381370225305702358695234531859564247230621836106178516852128113\ 309378231676615350405481774722365091744510701575822518510755202871908799\ 694427667428085010643043625365008500809715026517365821062560125350963908\ 341266618240586311926511550940534655954175847565113425926617800785383141\ 844488462665206755333759803904224421147511725526912243578770255946436247\ 791459282907987350041066380071073404898902229346689316291893942860499156\ 995345792869073065884461809182022921589790166320026160327145017697237473\ 591346174234588475105997576621946837562896767220633757395946377481764971\ 889581511921703414590696692791413048506792020503800536648891853542121128\ 671089002756511016348426805137801343438596831472413964261588063411940982\ 475123777356536333278835999011780024671576682784192090190438832119065629\ 851773717417992023924200071905290984469824147709401204614016001446943782\ 611226096977349417011321721122947709547806709707728059979413877105255939\ 424595530589681219555577830090903622056854539164677331938127063709027084\ 678901262989303375675385926850764963928887351526931489012529412008336340\ 375664897133585661829197261598341014043800655899570405488067587496161714\ 620217956050815648729371735824209865598536518665087543030835052313109813\ 909935388724006777681012808594027650920248291360727051564399616138047241\ 744197065730900274281942775580835985605003458168611407010360311772990452\ 398566570430862440144011865975812115705010637433491117745177210318115683\ 130193857859438443531488411215966308388786960189078530531012351884869642\ 119152283056659299558483392994146522633977658710625155888335811796752181\ 801238557498404306097656615418170087657765999254656600645859290377801040\ 544862625546827265147777704723798721209980293939094821134177502593007690\ 575513813607621718445779286038535938751291870233588054123973992618039336\ 644341079978721556843840167875394838259444381402037966830427176636196619\ 417547674745341177887428800546697741696974728277547740286426177367418431\ 766932084909383186416541586410874111924848646798424264150957608634552181\ 769879224502095979375814203345565704628124571098400396526917451094919227\ 981776713584433789235451172338499955593702736925408699397936031543944155\ 233339976350635726864982067741883243444227592683676263552116102609941560\ 895079463127998821720065985343386136192324928455324884838054818570023956\ 754322389912882466792681754537028776421654630611850217748832931361685251\ -649 1819692257008 10 The smallest empirical delta from, 100, to , 200, is 0.5874801091 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 10 10 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (210 + 20 110 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 110 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 10 (-10 + 110 ) (-11 + 110 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 110 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 10 (-10 + 110 ) (-11 + 110 )/ B(n) d(n) But , B1(n) = ---------, hence n 10 1/2 n B1(n), is of the order , (10 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 110 ln(210 + 20 110 ) + ln(- --------------------------------) 1/2 1/2 10 (-10 + 110 ) (-11 + 110 ) ------------------------------------------------------------ - 1 1/2 ln(210 + 20 110 ) + 1 - ln(10) That in floating-point is, 0.577800964 It follows that an irrationality measure for c is 1/2 2 ln(21 + 2 110 ) --------------------- 1/2 ln(21 + 2 110 ) - 1 that equals, approximately 2.730699778 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 11, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 11 that happens to be equal to, 22 ln(2) + 11 ln(3) - 11 ln(11), alias, 0.957125146885927427 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(23 + 4 33 ) --------------------, that equals, 2.7071714504536814573 1/2 ln(23 + 4 33 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 11 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 11 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 33 | |-----------------| | 1/2| \-1452 + 253 33 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 253 (2 n - 1) E(n - 1) 121 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -242 + 253 c and in Maple format E(n) = 253*(2*n-1)/n*E(n-1)-121*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -242+253*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 253 (2 n - 1) B(n - 1) 121 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 253 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 253*(2*n-1)/n*B(n-1)-121*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 253 and 253 (2 n - 1) A(n - 1) 121 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -242 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 253*(2*n-1)/n*A(n-1)-121*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -242 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2348277362883240295460855299984459891309698486270947568432222659296157250979\ 576789471541811056061829552365485414293763916453412828151624532285799176\ 106932656382638948647096017345964439990580709073022160738846739123159262\ 461789543757104161659975493472011575563215319012764106711685341700161865\ 621360396310940210414080647062107342590410679999597913800491269908012232\ / 5964530746744672071744123043830441146237941503434461 / 245346950764330\ / 285943322494237717027128686106899878815494101725103394751941689330131019\ 687429059257612568643486716157950197354955494307252246293331761399200885\ 561238515137068986886228024973292110229418080969658291714016171531603044\ 657686533152203023364762387264190672775242740226836794035179950245127620\ 908886717369133794747057910390219851263393033888318837686671853023853842\ 11290104164460524105353041748866666888000 and its differene from c is 0.14305430307091988637760793778027058220205441512468137993333172062158791178\ 477615813499603113353018916540627327438441872829598221110804345970160964\ 764474605487704605004150951345948975240006612371038832822650760491192539\ 079376347060724916316509306891237600653214676083095061531381022878969344\ 250744408689517738679301845351960660662843165679723892654225103000969485\ 386142969821718409656123853689767266589416893917606499163847415289524504\ 044638864709383660681123117950239947296094056389675128833573228478918585\ 628639168884490242003330501993477521656302503571029747978060009261772765\ 889422130236949069524432707901303347469475891374620916056377599128218406\ 706892514372895861461329489672350952917092172524152484602714692027455673\ 909758178469295114139215284445206035325025782993265911191047673201684055\ 287350506303293205039350785591840437774190613084427283101414357342636038\ 372755536153272064939015105386103060821417936915503912616334205597369766\ 036621203729644920357303622447381922993119451223911202509469643112384467\ 301347516978853100315842476136822596916912284915135772606459428033792170\ 271951140173595135899432914155790435307064337731472192805975667059977416\ 554526279861852005697815533291873119945696077142532188620769602282273044\ 673880608652798813884627555779707518393526604626431406654773394781021636\ 464773195037877824041596758694399692146756238077848815742378711194750934\ 028257699577155495389743829449986359818198724671628221126366154028009682\ 315660542815636234602349662745119462909131333059268252475006716890100309\ 311316487075488766379394202667381923427678454140384205587063837436609727\ 726079273879735956707211558386356740448268912287524449856917943168820421\ 412916403124292365670917584942184207655088220191627443773727711402704077\ 761537109127515681901072184722534880742564548893928470661820692328725404\ 336246054886286184677207698512393481516116796383127800559795267721181074\ 510455020376145642139534721789815630189192891748613998619511563484360338\ 779528945584936477113753896115504636603497259375468993160524976466465980\ 828944165957106808943973028597330708271346531942072018722902318018554118\ 496949793785613242902579655278025129509571969332724280863071305158919784\ 199120295867798214949374876405036971526429901116606937933720596990241146\ 449772539595800811206030044633038638191917212619406225920769800524243259\ 757119909849365556625078975744553447090035718534205159907255665070461126\ 360529937024269840467433253897599232241643565241461237282261927132935992\ 761330801015669560109112793575065513285622156866976540176532800493398264\ 224434975649835362611580388047613245842990188705984965161270535073357196\ 327350299032341847021273794748975478436635020248691667700135857421152379\ 550467708251040124288884212040276794894342044882036975864345949622406858\ 992836060835817054675931825296629698568141242579642710373001581060234509\ 319730955575624105061351673120786242368595614953504169835026217936617077\ 772464310383394582693465008837952501224901076535264565468597851069636956\ 833665361794785694755384965383910007350758699678706836826424013958942301\ 502262410365331942405284644750596181310888885065096688783717706381826920\ 768259823794113762945734539720991813742430212046752361429511380907451110\ 001510508587715066647012144353485887991016887233960325490919435185061851\ 051501579436441062277681645738275415268905369388179685477386923679492954\ 498098590224590316302541352474313206918445866381304833448657667148859347\ 469857493757898914511563750789541338045745853147176676356353127927949471\ 812597215671531678365284756816348475639806676507366620009229432110872584\ 311494322614164810910647610716765996872708929339737807416321871420183968\ 454836062008780686341355447009362561460953487610485404496834145187199778\ 161258367888613821045031782546501103175898374544707211555122484635802511\ 074642856053660380889211482420394588113210412151923064896217064188210020\ 657758354070733848667288721944203871036381711043341825738855520617095638\ 924591947084469367888690668006971986913344133606154061859737064411793104\ 406174009860720499698738677921394040255035870055036656351857760504140624\ 139827982462756078450298539680661653983723174637235645774604484667203671\ 204701792445660450469023013059235608030932381781042389357222786344819538\ 271912300547691994819573117143883387856434449194234710917830854859736823\ 160680475926083680735503355702182745898567782779979705790392366339359777\ 599833519993864003597976170537685756152384060252806261090419854689550900\ 781142424214219704786379197413494306668267600007416797934359866412552982\ 439147038725422301622783830274060905085197713327741809309931628306674893\ 798922821022501148572653189346060762879491489787391809866616505711588951\ 382387269719059866608819538613418803716794737490914114670822078245006289\ 811697521668096371003162731531108891003880965960590576692593002752302968\ 372370131636888490966371612965491706639498087474300667829600106713917804\ 638247395817889779364579789550650330351022527622889636865732197189056825\ 058986875458997680052412022020467369045797803666602993050402840261722331\ 420992134902714285309530936290287766250739705221350953952366951347510707\ 916716578658132425165379593201799768848385568896167328360428928793778131\ 686645114117385862009568025066628153629335092962378283909404350042545079\ 730471340648251591169523626921668645764296027379860262555900839624559932\ 426143412566370396712797215848658394996304817753639634378995931924167318\ 646909620476408203927348239934098312275375272883081061160674649278151769\ 811616018347807353638478837826380532932466409081995279312723562056095768\ 681487652043288704922899985622353502129168106816925256712192796931439415\ 196845240976089150767119931778189316188427558345939860698882296630023301\ 294643670498806435386610537724689481488955753090367608344571106977954693\ 886186105623102284119506121466493733792569950853339812021827277644684767\ 103071274370329005767722262066011816598024303282315215774799929197828443\ 148179758113792342302769407827947589328102515411224172512426296824940071\ 398756542052840126503948869817819454360007000114399493563366372602987645\ 165823429386769539634852432724213648881736667970043552984434659237523939\ 774847236135347865795583962679447994952701829103551250089880070400685814\ 948010206047909053313915858630512394753240056905606504724592297756742822\ 573891887331391515168730146463911636316678343798387353710811097003030614\ 6388973666856904574004722572324189362667518453091230760668063909148717 -664 10 The smallest empirical delta from, 100, to , 200, is 0.5989222212 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 11 11 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (253 + 44 33 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |33 (3 + 2 2 )| |------------------| | 1/2 | \-1452 + 253 33 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |33 (3 + 2 2 )| |------------------| | 1/2 | \-1452 + 253 33 / B(n) d(n) But , B1(n) = ---------, hence n 11 1/2 n B1(n), is of the order , (11 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 33 ln(253 + 44 33 ) + ln(-----------------) 1/2 -1452 + 253 33 where delta equals, ------------------------------------------ - 1 1/2 ln(253 + 44 33 ) + 1 - ln(11) That in floating-point is, 0.585764183 It follows that an irrationality measure for c is 1/2 2 ln(23 + 4 33 ) -------------------- 1/2 ln(23 + 4 33 ) - 1 that equals, approximately 2.707171638 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 12, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 12 that happens to be equal to, 12 ln(13) - 24 ln(2) - 12 ln(3), alias, 0.960512492082437110 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(25 + 4 39 ) --------------------, that equals, 2.6869021714949390619 1/2 ln(25 + 4 39 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 12 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 12 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 39 | |-----------------| | 1/2| \-1872 + 300 39 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 300 (2 n - 1) E(n - 1) 144 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -288 + 300 c and in Maple format E(n) = 300*(2*n-1)/n*E(n-1)-144*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -288+300*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 300 (2 n - 1) B(n - 1) 144 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 300 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 300*(2*n-1)/n*B(n-1)-144*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 300 and 300 (2 n - 1) A(n - 1) 144 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -288 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 300*(2*n-1)/n*A(n-1)-144*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -288 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1832391719153338141626710794811799938737638575965689245638492543523824979320\ 460101276379091738626675714107830672761730683242336441501158615922508114\ 018647001737628367940854651798896504412419196694199252827909345954645060\ 504693715222087867456025552921560081311403671730735722118509479117601717\ 721063357751554881015096096700755710311669615500410855621600697623737428\ / 32697832630480366498603285975124742907431890177757115640113 / 19077229\ / 440094267672060946597275194678172394581942019948275040089291227007115195\ 528696391169476043799343790433965941085902687759502890879213084475358001\ 612217294386547050726164834757164276652148459823977285547121365867975807\ 266166004793883890007036215740986479716476169105355212686445766896895410\ 735704899897575732164687919608877545785961544484520713061043989712834896\ 7522535588592924591367457539144141863703672740371432048 and its differene from c is 0.45661491862613924266900109702583898420194935466642669583308377955115439488\ 037333934323387099632629598802975788867605018139795083754406022325648420\ 952023162219662298036337323727412528811274198198008815972958069497656277\ 776177553926582663057134109676088228875427765911519931101896240739857663\ 128802655886005313914802493436339780966933330445585873803371854421122582\ 332025306990778080783723119719675237204542599899922509909826886878262335\ 304937119511311893291303140606672853429806469861103721663100895202700953\ 890924449039704699978858256351729092276271054789720300089995453727840445\ 393316905340503705458075935680846776543837480773432754783061030799603910\ 753598742961938519700466901164426900613795754388988719311110101151034934\ 852420460375333943388635940905314926577760036531880267687330872742390345\ 254092328702331446314818339722879800566350275897449229855877848474311790\ 067119686077733895876871409415053090695926602544694051360129640025639418\ 319440640869488832566598507726822759436266562598811642093607135869870371\ 550731492241006457947635448214789797792870964718003594332105170405462078\ 481546742358463137103830794158613261519870682118453237863903318167965675\ 351277833862087773953941651352993159573272165679786564168413250498828956\ 850176695935882119338467691508478585608946047618123056100038577656783965\ 572741279353700207930914708965189066135146287601075040976958839196784769\ 996123089760347812827423122129276987649916713383421083584132713724241482\ 701559058195823359422628980085183386807397826254676389859311247117826905\ 785265085326640658481851723490671272079894079346068469681705300210889405\ 383431326856610023814720307407489681411119140345141761899551887786922160\ 981447058553208260908415574840795187818023252517053702295234964819086056\ 449817178944506700686720828665524311249989842859190348210124964322827400\ 191041279879421786225217102709386264321037569922722793960058904101436490\ 939019977688828664525392073555904750612112126411782259121690499951174609\ 747583925123834485591260551532276556826900711987759316561045780312614103\ 704190020095886515909284335894519574345794744975610676484273084072244277\ 035774604024315607940478340178249242461449658531732514466544455762940820\ 209033828780535767993183259452532957402634608939180916395931626940945878\ 535293538293033457749053424920881553544957812021877750575702245158501052\ 686835751568164785554879562929367804767902012360016044699077806397579409\ 760629513215891585095110197103170647099956166687359464825203126274412286\ 085178649586051212275356756533553596677559138358114022915331375965204585\ 667989442512466981812716433364348527520717168858141175849710885796358991\ 118887930531544637993687798996381181690842652184929232873000254397233706\ 133218039890432163044428926861796854040406016673469442369661983734265162\ 639650295152176633162333826925535579487466932866668418046771260435028171\ 859755141890312231722722312584166027019799787768057283788089384053379276\ 128942069733929388886071861617662543235462184025402808605693037554923215\ 789464571571463698445405869969461795285694277650397819788674432408477363\ 412653356797118979815941133402070946953013773483621411171149131742529541\ 054112788119019711191343959173381765672827894563287080409411760248808187\ 954020324409534711133849593185932887090262565206399873045474512828888092\ 413737339795894014463766025090649539750819501525213892129826332395927492\ 643002717729907981186985903240871586409532261538552336235461555785878968\ 243645499648580795828953199464308207416430596594846497066949207396580393\ 345693505467339616578315694247211073128141467979619667797456338041264616\ 462493092679152785422613285284485463995404284969492994541403573213461338\ 223537721224683248793429672448564902018598483853416526522468315897058176\ 626063656442502481903932446265247948897026320617789418866175751651388334\ 719238663398489676735389895433742663231423220722716595607810746853707298\ 866955342374454393246967221699345133357536350939693224183204876498616756\ 974966805258334788957916207605336017741081659313793797164264463369004595\ 981108508400229786627704218765996577696197077148936670580248236404398410\ 049650195904066578551444348765552240515657362576098816876998889657229724\ 722133837714994012177511454580715763245969960508616078981281507129242726\ 399925768546039170240288963196702240678238475608321765712841947796548907\ 468532007446806734369522647414747495911292841742116485499853603755483594\ 592034378207043362505091676618716706438519592549303175759853614220435810\ 082412330372297154479594874277385344889511235178990620407590793480030008\ 844365643341615656110013189798354038009875851412151903839994502442262449\ 734071975902763283617257294076420327931295604833296265185627398157775759\ 094675386031659446249282765444313057389056108785816733024345765856322707\ 140676225606947714259686468580087583602646752884823866784514468545965977\ 008215721320198522383506740028225236608081380542632320108697786517682760\ 416041093867028065574009253556277847166007027198517492493283188077263277\ 051788782729213154940685050514813216807987452933160068942836000280634561\ 671482742875462709347048950616982328492977639605253577883499479194322988\ 948360040331033390867967855533125035151906434648796660778252473958058147\ 958587948321873531563014723826758679552992900644801414075932656585365550\ 781647700276773002331629877751686096352152835321023382586402302618244236\ 926974435009303942868305810018346380159629684555573642377549265383497220\ 295756541439574260560097038849109015598109701884932600483008091530284653\ 757813178183220693716828238214884332502692134090535819020774113802892019\ 613920851220312398349868081551097817034222535284455062448662592008105027\ 385719615639940450366873426107706576047610309507439554113668567193890532\ 897306117999063768187903077921148451625741184944866495204869748105175543\ 655827748459414861903242506454368539338121424216806882685515384837536922\ 106173642659478556007863929191920779820936266715947763757528292933921276\ 691279110695962773685218993724221218567771969881183970558714138011333807\ 489580293154225022110282956610704611370178245205764179596345347412028179\ 177122731846972375283027684162668167960748833409106066552268797442816945\ 026106341293309971645632765653900021411739470209069497949346794801352995\ 573638589774913818108187908986298409487409650460799424252019125536524806\ 750266072099898984425912510432139684415116221475384844513898252320779240\ -679 4481341573563083310469710941552164680868618629667818323 10 The smallest empirical delta from, 100, to , 200, is 0.6041656624 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 12 12 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (300 + 48 39 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |39 (3 + 2 2 )| |------------------| | 1/2 | \-1872 + 300 39 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |39 (3 + 2 2 )| |------------------| | 1/2 | \-1872 + 300 39 / B(n) d(n) But , B1(n) = ---------, hence n 12 1/2 n B1(n), is of the order , (12 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 39 ln(300 + 48 39 ) + ln(-----------------) 1/2 -1872 + 300 39 where delta equals, ------------------------------------------ - 1 1/2 ln(300 + 48 39 ) + 1 - ln(12) That in floating-point is, 0.592802675 It follows that an irrationality measure for c is 1/2 2 ln(25 + 4 39 ) -------------------- 1/2 ln(25 + 4 39 ) - 1 that equals, approximately 2.686901970 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 13, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 13 that happens to be equal to, 13 ln(2) + 13 ln(7) - 13 ln(13), alias, 0.963403637998384419 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(27 + 2 182 ) ---------------------, that equals, 2.6692004974288845362 1/2 ln(27 + 2 182 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 13 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 13 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 182 | |- --------------------------------| | 1/2 1/2 | \ 13 (-13 + 182 ) (-14 + 182 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 351 (2 n - 1) E(n - 1) 169 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -338 + 351 c and in Maple format E(n) = 351*(2*n-1)/n*E(n-1)-169*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -338+351*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 351 (2 n - 1) B(n - 1) 169 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 351 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 351*(2*n-1)/n*B(n-1)-169*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 351 and 351 (2 n - 1) A(n - 1) 169 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -338 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 351*(2*n-1)/n*A(n-1)-169*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -338 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 5766333093538367148798430779959418392289242360536888503641608740879228927360\ 605621150155505617040506181568221372324416510679254557521915686565588702\ 202105351816837502321092101326122992835243809174028930258869357046351638\ 313288451825052471890920547474358779561327255123552822682239262479705925\ 945379188041944950612262778212051424702392583025249444728201969106162943\ / 8833973697451765392586187604329817404682520271521360962418896963879 / / 598537608340237241464837815541209037903667974970094699002093989257630897\ 455973784533188740125262837864873923030799505548055389304161792809835320\ 085365804745413813304197189310268456994722706931990533199098824426758437\ 616811532007745322059300786326314220416311057706583066556755543607046325\ 397846864596168190364324865240728297675473449898142891440747801264540273\ 87524330753791986906292451672525586167134740496932487872431379040296000 and its differene from c is 0.19117774260314660095438209939340597591715536532283364046614224772758134340\ 366728187723652603866350040784810881976131418155495190434690571304673046\ 421667922649683462063195528589960388947878356589426715912062464602984965\ 702228912873155325627347754802141805592679155079010142497721554422837004\ 551991880480655511021338328017839105030107334829585398014965984039524252\ 152146330163703390871550406924332859174467075171455395847162835573570474\ 985598492167964108461636080532169237809138230489914516414713043383274359\ 097516374271065138495859557837911111751987167053995675298256365992762649\ 418663350261320603623773149796493938142904002395301808059840738049912139\ 057548723441638951227594969333778403907455149090991893469822196539936441\ 537410200441707156543516688971507004500713647820032753361268683311824319\ 046941797783433985582538919866182196672720870967147706889561848350371437\ 450440182113559015049763625517303867802037644021559601140434482607369912\ 937624860539360440426803178871900050834252869323947281708891895491355784\ 466290489374416529484080380710324810837889581263779810470694526886099553\ 451645814416624870367053239101121058685851539658103644174031364814213537\ 348636563995075894957369606761688283734285306189747613619246888974787561\ 071101237808134004615742762236121060139942902169348432418644703413351214\ 961924803031775341113230158293521546430926476948185214122302895340765325\ 853824739524120845250435146187491750679165776265425765525476226368937634\ 767472984899911451821518117562994701926496141267844509440982327633523879\ 133607184871393783707930669044394876913205814215490559868688313844698292\ 530933469961253229223915742899677852944011239111738610824881673546714572\ 355202604262215971600826900723934367679500458099325964991181424174699984\ 932743269844786139402144300168377391491190493041516451686561852870279196\ 393512662995335662486521466898687382543856004449647552425408729180024983\ 725514501809192407147673547636853001067957503117089126774185332323955285\ 265258150677893318101308138314044105688762683029711919695156045290306638\ 128105678975815947687606909589980693329150232970547984159414497492820034\ 175841673314047460321662812728665753603057163900093507285292955500470839\ 710220255718437485364978317368862556724936324809946401122646663056388716\ 447643252502017913973571622953534226111006409917566573891950404090871273\ 009924683451386818304334512574659836543947088220053801067446626684189714\ 185614373043849751564790713428515725496420769386062654926033745436801652\ 714480314224797162176277847772852229493733061586160413931509917022774113\ 977550902863813879552280255488171011875760628687054072024100982286017769\ 014885555577840579987059823805016086123115010675165401918360399530860194\ 523307946624471118338165135387204523383389401209213373084961725359814557\ 942994400575264607760620362997363622246873544285437707525687703533214083\ 540114847533361752089492084068912152627300797431705108421539112670405138\ 412127794466339119843595314164218644320010546656305051434286295667354050\ 434396966504884426805634350104331608228267209619006694354506394751686006\ 227468760134405349876809217883903638398915822957905571379578945446629233\ 534334424837892672804607185964820644774107152791588986889230789793798652\ 824324529147664786674102434274888050098778453494958351029441133923150292\ 809321209972020336230562130629866464379065079198418001105996831606114735\ 846440828615376675642540684909879061239426383363213958814598494690305569\ 045257541478118592539703897737203279883752837753481060681068863265007058\ 573114927152843908420564693384144436072211011456555319510159940971662876\ 808654659779172075764993220570854000444858041085202496789318433985425204\ 118663267154374899865652377970405367839166024371528401104115404666723800\ 709287212052603534711922399072007905413433907873524340882615827900700193\ 166306755951921835941650611242938460548263976837502551826499182475007387\ 920246403090785809150800002103677360235217090989252014398590327959195272\ 584419282464598200923206267336043486512186417945871968058487826307828523\ 666448651849614921128520022817936361778955959409799276263155731206975563\ 215140372277892969319016205488352138131104242637708764717642746911230378\ 987738312371864138241000241700849468645874721816543136689847244901835588\ 021911760751458019735595058986599013740584281347042447225169036467093134\ 429570757937584401160870220558510309877219020979080663243969519310641781\ 937730234872558270598539371142052779706891356336801735988037791866103673\ 213756114268640577821006970696610849889683824698727454425072902841488628\ 472793815151071733826875057191564743727325804302307938344337504565544101\ 547049867432925020380233759455240231484567751862531458313457243570792158\ 386572590126853536509516681756101974154808554713832623738757229920528289\ 578947544902142163653424400945281971617609551266634980020216098390432317\ 634678669658117326027595807909606614214161710986459252611403852422987075\ 377049237644826992273500774282373085632271837854953950656031801748172650\ 875247537116022810284294585777495385364428306599560099739025082954695317\ 924983871545660387474032482490659334477973980773963890152405735943382175\ 816584554254904780774737828848907755043592327113448078940885582766073142\ 885755176661109627760020289910610132222709627889530967133737076772408954\ 213479111245474876317131409813162266628951968088101690813888024770829353\ 285420672650755365348226090825422603124123638849785143195608581893357230\ 191855029452612182502509865792214926546662641953918122203385420973552356\ 269973313214055546035217152838643888706355290830935201011239433238661657\ 950941382104717169084635083945718341209006577604499958508124784013235278\ 344614286405104046703439185325854322050247829845234439896116448971953120\ 227203007814729013576932645601683343882044655949731412174960112571816200\ 594856346941060409997265131964678673480216618782154497125348968635920032\ 581691280312193904134440816495555130226068665867705439214119394673440359\ 452665427205934149390756620975240078493304076415736126611238418302752308\ 432852975527759001552429339007040794903758941009103738439198993127732892\ 808808453285086214450986347856066725705720281183587175265996545157026099\ 694628792197861825891280057157617489306074964760074069515086939732487280\ 538877591546403583863157448277164509426991158118500182461870956786089214\ 444131357315480700199052247617408824058054835021131933634212188554799423\ -692 615385604287090393675225888042310864699145 10 The smallest empirical delta from, 100, to , 200, is 0.6080673193 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 13 13 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (351 + 26 182 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 182 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 13 (-13 + 182 ) (-14 + 182 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 182 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 13 (-13 + 182 ) (-14 + 182 )/ B(n) d(n) But , B1(n) = ---------, hence n 13 1/2 n B1(n), is of the order , (13 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 182 ln(351 + 26 182 ) + ln(- --------------------------------) 1/2 1/2 13 (-13 + 182 ) (-14 + 182 ) ------------------------------------------------------------ - 1 1/2 ln(351 + 26 182 ) + 1 - ln(13) That in floating-point is, 0.599089205 It follows that an irrationality measure for c is 1/2 2 ln(27 + 2 182 ) --------------------- 1/2 ln(27 + 2 182 ) - 1 that equals, approximately 2.669200499 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 14, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 14 that happens to be equal to, 14 ln(3) + 14 ln(5) - 14 ln(2) - 14 ln(7), alias, 0.965900200817320321 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(29 + 2 210 ) ---------------------, that equals, 2.6535636708488990746 1/2 ln(29 + 2 210 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 14 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 14 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 210 | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 406 (2 n - 1) E(n - 1) 196 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -392 + 406 c and in Maple format E(n) = 406*(2*n-1)/n*E(n-1)-196*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -392+406*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 406 (2 n - 1) B(n - 1) 196 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 406 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 406*(2*n-1)/n*B(n-1)-196*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 406 and 406 (2 n - 1) A(n - 1) 196 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -392 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 406*(2*n-1)/n*A(n-1)-196*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -392 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 9963257877615799684034333782402599745317471869475994593967412372643454531552\ 004015930809715118856020327282033584496310025023171684441443711946230073\ 510384982133413525045270577542351150939215392711914509036875243913337594\ 782638409675290475642258733799099868689396079995058175720174791867561359\ 148367525164443296877833730671179098853450117517483554158066466559961904\ / 601399536284968693922690286344276828633963068159448914309079382818169 / / 103149972110836532140647624951693930036810619190588061195816891743518413\ 841238279013989860151691135996013533422406803642693365893723477970776804\ 904119484771533295284044130391711436883043598810324051587024170177837206\ 474907952093177402789819559814819273560271710752507749694989034833308248\ 851959929779088179613253502842162012253811881520824221562611593131497910\ 088686131212727207011739237631065733660035072506355717573916685115001240\ 00 and its differene from c is 0.72599410881263570396293235893001611384800639262835660369794645094039180587\ 872894558843912301274217141185164531631970961216363773978056979059381569\ 303580248291102618912666545986998745380630924051051245384660198757856073\ 991923619061814136723748349404359083041363947793531990946185694084748945\ 638877154970953577351529315047718373293550451035586514413363471883857127\ 473061310760332108734632555050669071784965713792324398142290095912731752\ 317431896279358332863005544919445274832096860438119827485557285157860290\ 185657202606978545140328586067255478312289830237202357528700751682765347\ 081063188488787378542363715556282211841956005309306453690965261940340303\ 096086418577493965760220656011994986400318816173271530562617438558465528\ 490722942239403604555831537361854192164010503636165911587259777011072227\ 096201780925137833054678127190679823474626253800587289802060216118883382\ 065080424207415408485864648506150195943244917488387604940542404016934707\ 776560618489261446746623540399092462248721297233455129821974239618449608\ 342418835143679672655405526408181233153788093726291745719975143694584001\ 508566067344644647197396161217713795647667617564109891196195416073999974\ 877683125155041674376029870106504985126279360422415784946341879381188912\ 983583490728265139487782250432563673894548964280874011479179372199478896\ 897980321597884536017646345988441537357212329208381377812283861681818313\ 565879995606629182670171857315989686227864631064083875729365710576526735\ 906801768733607229888515453104017950698925170203151409676444659980151198\ 338318931052913931431532771593351110603157524650731678352056337017427083\ 911542347251325047879695359455498112124302959404749842982329932972791355\ 625103957251537453151512111144150788264859326339099837392575390250660781\ 412505314233265459073005462470088171640158654574132750938925761146618662\ 305552243895438129927722397124284081862608126448289400356846736954190823\ 253323985669396408924774455017940939583856318808591775496487722381324063\ 365916652635241192129574104623042819055671259314535028952835316338474231\ 537409551126958935702754627387791566731376216941217924965388356325075330\ 413280705466146854750707566250016909069003616190119368864988961006789225\ 076339943379014065920907995567706436622355614755637160706514035496279944\ 763335544104974384525782110242953796099792403070781538020239717868097013\ 096299422175786930791119199422347858250151080015891415079203201495027628\ 812479444207318765435591012660681882209371459875158171385775388978185383\ 305821875328792184852652813070404660118880495635187553298069121834877843\ 843257038740833483646926145509198062801432083354832324235866371035093208\ 293585809863618936391900041342632098696034731659296643419198343048524040\ 493929096355521127284763582068592119511477604538039425603832875613889269\ 090479476173253956441974116874311509278329537376293327070887865574370115\ 772358779614051707581966419303950515969154738754228202634317196229736522\ 582835006937907146095247040217132086505859714672513892064910348353025630\ 981400084045044464585601471296453762949506714261915756911873700648449812\ 478736193870833978675598181287623973030982944851261984014523926245946211\ 229294563026625210942866550347491764603287468847704627515614782046026262\ 641444226759449792272478676029096001853773146822781841307742634020527332\ 129757757716138339113950692489588662731001992091377882429078078466756545\ 718532710549898538622126609304985966422871241798542738304340889273576487\ 353998272028084016763343689935524677882119060696535891742562566218282094\ 773928632204845528631918518600984651621779219570301043540136248364949516\ 224827723343947222534906137073152944471894097859301912938175313451884342\ 132543254868328040690055295722515829344635080626581780290276289238108711\ 125228870136247644023855405228897802145381854052961985234833047822756612\ 540926484209363770281050932260121616308404734688846697117662938265050309\ 338764808839909556655984068991373051660937573913662064764799242783353342\ 688898767984376937339294709214703232898478954411487420792163030678483589\ 854961757190576096676983183217044805950641117533679500403986342166467559\ 396587607436874209945021291802135114988304833186435505790405360134507065\ 586002961647158536095558719375654743799195093736388220410331631867945600\ 168422177360239948156403948285313678518134710607405744644976881910118065\ 025390728676299321495366905805090643387695560715255552672294364033734473\ 099989360528597783504847908947722273964058155068351856027529388878264888\ 101163263456763013881529033330623328863458654089723703980757281238190772\ 558363417486198228436143342036434942040083354693755397867708627441572836\ 433136737703562991886891049946333793116395624856673246162077868104430120\ 534370425603453567163553568229824055925963330781116791394258970847642154\ 241344352081714066241536503006965854041237092474173317976952741793929208\ 809297073582428585088953024759488143407396095249955181040374306267988986\ 229040521641497977290994308918410298818859249161356802134216471268911667\ 869965898115752324209699150197152432701709967538339475443108260604226187\ 234409880912814073680854619814880181636746666241042907870545495730618380\ 259659307431305020520417026855765584246730328056850544258740804072769855\ 108730945839981892060928162768523841892622786535948683441054892383216820\ 351631321899634661123151218358091560881393817325393569546721230895891905\ 034519054833795451720397206344040370278711551021458038065121103967580787\ 161997325785923960981835631390484975616886050646034767044511294755981619\ 284076692034535390694153426685935770560767119647182166750484877955988949\ 173944601950037455227433018438300352008172335512305743068632852856760604\ 385006461742521287220839133487136004493768484901046552795868034831859946\ 819455717759547880035897600816160853221957261776273077369980814980922312\ 507230128920046194398687503545029274018491800738343776515767339110412927\ 568824599233826159165761755758658725509679943012928910762624834381680447\ 917790575643913146447417429159564421468316279989148981810436096542158404\ 148207315245113352591783544168471940309313476336006810616594965776715892\ 331915386559483696942603992360387093410115979162579084665763754301957046\ 945202799317652180545104903949115782234437142841032954759836952783776977\ 343442089859473242931308663133360421554239271491627213357879447570998262\ 411118688544709645116507480129988358035578011543894365574342702902851528\ -705 83742589984666703159325689788 10 The smallest empirical delta from, 100, to , 200, is 0.6163357795 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 14 14 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (406 + 28 210 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 210 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 210 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ B(n) d(n) But , B1(n) = ---------, hence n 14 1/2 n B1(n), is of the order , (14 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 210 ln(406 + 28 210 ) + ln(- --------------------------------) 1/2 1/2 14 (-14 + 210 ) (-15 + 210 ) ------------------------------------------------------------ - 1 1/2 ln(406 + 28 210 ) + 1 - ln(14) That in floating-point is, 0.604754457 It follows that an irrationality measure for c is 1/2 2 ln(29 + 2 210 ) --------------------- 1/2 ln(29 + 2 210 ) - 1 that equals, approximately 2.653563671 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 15, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 15 that happens to be equal to, 60 ln(2) - 15 ln(3) - 15 ln(5), alias, 0.968077817063567575 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(31 + 8 15 ) --------------------, that equals, 2.6396164066305640222 1/2 ln(31 + 8 15 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 15 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 15 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 15 | |-----------------| | 1/2| \-1800 + 465 15 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 465 (2 n - 1) E(n - 1) 225 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -450 + 465 c and in Maple format E(n) = 465*(2*n-1)/n*E(n-1)-225*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -450+465*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 465 (2 n - 1) B(n - 1) 225 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 465 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 465*(2*n-1)/n*B(n-1)-225*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 465 and 465 (2 n - 1) A(n - 1) 225 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -450 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 465*(2*n-1)/n*A(n-1)-225*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -450 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1346193123839050787880258483951440355601293061648132047670172549955328780493\ 172150591886144746011433176995568078451862806887488313852219366072545114\ 446422307735903589444971397034271211775291934046077869824299702599389240\ 336825444697238474164412502396086547290675754288056860755073981318643181\ 750822417288795072213481962933228265276562108845542143496949643247036055\ 468829414787341381534475644547488927983340680759535663175681525859679217\ / 826563 / 1390583587507877748692452463533668906836237986672869477152481\ / 559105200919967249559630033909681083650655513122414290304409217911424144\ 203438071788083745577474570128498621196426648991207553792829867341100333\ 573199530145628206956579739705299992456014580136647514200919083679443833\ 876442377784844940435197236325897372030219346908056034434219843358864508\ 738356792508618751453499165281763409438516997759933820899691260345358182\ 853169922224091275200 and its differene from c is 0.18620204455834639678539078380489451244862337869714568307881660754391176922\ 856032641723046356586961199239900593627185555902920294921702427649998173\ 804774921069435694731704294013698972782796320414530022809005311901149142\ 625420365370001758308945841872498149083256643115263215130842112448922798\ 007810864118283443726577964273278307445614389070840447484796371650117550\ 774199970403971923140974812589322338186953703399813907006377258849303647\ 713180393659416460821476167349380747037698205449998786072298638874895067\ 721783395156822257038916193826709271416270648508666280670133794319021547\ 144321485102755216846280318272991328924795183958474519800227917029197962\ 393726613127724352863138203402544227382742136689570399466175877251309616\ 148798319958693485999509119433234645764858214809687657992996784401024094\ 882874676392539514260255472931165060654633653490241428758079889820133542\ 805490327499759163800745378165122149744997657003775898981548900570710475\ 913420927437884759438448356106498695346578954761346742858034695358754222\ 358979164416115109860267951616855135865327023253924789077949489728748379\ 150106306020227585336729181995608551779970337522661704231434431855903232\ 432275639987477777300943203659635289387208294884451263348607710378464596\ 868892926991299266759817020782569936952919579276546596426905877120217160\ 260624884392043214729254309934173877439897590381803410420008216414217395\ 084068392424860070844169249923754728272113508111545821087488208733560898\ 986608854768581186324763957552668653379180687392378367464813247631136058\ 204498262123393835887890028781244410549856643788181346212689958317195704\ 142119748512221731797029950025311270348577931172475141189463561216448237\ 323089084238559003930606742941695296572666361013096105503801370520254751\ 058800881204900648618638388940481835747911791406434364382560859791594498\ 011490169414575493807632941760851037576420764060739009988032475956664009\ 487635299885241296283502419931467832427366489712135437578690019066887059\ 208881467669773590136228233891137882023608882951811338318511800352577449\ 853584034278482692607360689557087666221842276626183373269557128310824065\ 827998285995999772481118768468336627111740795586345954575120670048754072\ 281027755155710158786758497318464121791367292169399483447883647649908203\ 695549716449113790245741222854226831120283940715433596745463662805504107\ 355970876209987020470729962159172806334016246657403052974436397269694959\ 491224989246166454975606766488097611911198312661632873729736667585640011\ 822367537167330566525909587935021707181324336830289732674897346610841689\ 463645552788252438781092974901381448796216262071441798007873487460564694\ 388140683800650391104972115104361489891520369141705000548996467881398650\ 043946962493513460948938259817788405015996630819350612624391118925342232\ 658728817717299911935910471173654883325223237043148927962383198184340455\ 177368901260288082092458130857488824282251607169384155040394869857458587\ 606582335769954836406191332374339754323910986770832764165217975773663277\ 579680243825317357793012691225054070652835093968338987349991798282406559\ 376920202107401485492134650994461108422469762076004861901584050134017964\ 514849389939724398610341320105281555792250348942806734455675548968446231\ 539711852579596654429505569806512106093465472145856755768382907333200426\ 250844186936098237255815920810408120437130916364745439889387105645069794\ 901331548363121366456422369741438565247017523145024007171277822382072072\ 680023231463574035788182295286529696334946587380077056880886004645741829\ 344034832521786681554051694627654826437384063582167217728415708850624205\ 441638425092408462148448362662666722576974101062887528994200710275033021\ 456432532199422010362824387757921201808352071535260520380198053805420656\ 579400357178665057566131522835131847720017346649255949470480464922453731\ 301243866026543064502170698131737766395598737709086551753931924141406257\ 061544056914744861011655384569563566541026680916198487465472082203875250\ 310412950943433660756571559946170176104581517046161241545360521711335849\ 632165414349818258467786979672592271511554999014806639611435276563888789\ 311452511572031510788688322327957814119161242822259262369534542768467031\ 375236405100065647731014704486242148428527390924338008183307237974964083\ 162958064253503316166184538245595058206988288340984197894724236432821033\ 889406454076309639234469989886538566886712771371648620738544364185211497\ 256408452857946469975120086326894564002276437662668152133226159775641329\ 500293231472948743691709622303142359186377579560664853443765327148018730\ 979081054287569828716585754480447858189703919941049124660071307138467055\ 303104811671390213523152347120804876955865307183050795348441194563934849\ 488760339444799162578771047661539062643173943882129274202750479152529493\ 345806352175432431424093170786998848698189191543353638213436418793936876\ 605290160033245668649397297294403467880705176778615395809488692099222002\ 436230818055886180633307329799367167844456384944118565522240658674058361\ 481616779692758086257357455554716617451052618320704617455051726083436647\ 140728666154145424099256487382673107775615483416437863831010577047490383\ 548365272879440296241030845595830673364674598783555141108127350212225913\ 556249244040614257583869659695811184395843019756868126573168181227293223\ 093840166290200488079834698923159621987716765870123095292660985282928787\ 915818840202987643777791568232834760968302540973516546994585474011972803\ 379107029043211519150809171652555971702657937863559360068345035642778844\ 668745742291191037316883154308733506015687657701752376922661194431296583\ 512959855679792108013789136410885025624582283973314130688581613724016920\ 576924422280219171529807976706847711931056865733138910680529977420649341\ 410699858727425800896042625800110949024354226384003048887499308314662130\ 524115404958564763650772209315656031744886957694459298587205685053679305\ 858434655282683832797876809118109555482278365557196818271288635225035801\ 660834452383715538288212993527538587131124980660661804231152280467538207\ 587068269338770549635478644193074590297484083676268971819190729786899099\ 913783487813874020903303583317990701291912920481098091007213314689864089\ 669859283632342758707228226254022472454859221794425992958357595192963467\ 133780359482162886376384121782758874309015543655257299801065355133433453\ 350938084325600264208758069788842816938493536281924446684332068510966076\ -716 025318289197911627 10 The smallest empirical delta from, 100, to , 200, is 0.6191577652 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 15 15 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (465 + 120 15 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |15 (3 + 2 2 )| |------------------| | 1/2 | \-1800 + 465 15 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |15 (3 + 2 2 )| |------------------| | 1/2 | \-1800 + 465 15 / B(n) d(n) But , B1(n) = ---------, hence n 15 1/2 n B1(n), is of the order , (15 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 15 ln(465 + 120 15 ) + ln(-----------------) 1/2 -1800 + 465 15 where delta equals, ------------------------------------------- - 1 1/2 ln(465 + 120 15 ) + 1 - ln(15) That in floating-point is, 0.609898749 It follows that an irrationality measure for c is 1/2 2 ln(31 + 8 15 ) -------------------- 1/2 ln(31 + 8 15 ) - 1 that equals, approximately 2.639616414 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 16, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 16 that happens to be equal to, 16 ln(17) - 64 ln(2), alias, 0.969993949062957480 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(33 + 8 17 ) --------------------, that equals, 2.6270722593344408025 1/2 ln(33 + 8 17 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 16 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 16 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 17 | |-----------------| | 1/2| \-2176 + 528 17 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 528 (2 n - 1) E(n - 1) 256 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -512 + 528 c and in Maple format E(n) = 528*(2*n-1)/n*E(n-1)-256*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -512+528*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 528 (2 n - 1) B(n - 1) 256 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 528 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 528*(2*n-1)/n*B(n-1)-256*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 528 and 528 (2 n - 1) A(n - 1) 256 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -512 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 528*(2*n-1)/n*A(n-1)-256*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -512 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3220220121571120208737575142004201144739797877753482829035810054100678967136\ 054431290377164714294260436488189923898946877660927548864757229186621629\ 431177881133642283433610780696164807049485289742178530444776503547350649\ 807328638123107778435330046920255662781800691759791697982445201102334187\ 659455306576193272198545334840056129180068691884027529322719229273490687\ 084360495865845081406723154192608418848752173855215422612096998505461492\ / 15103432003 / 33198352677168213365683721785476908236422706466069514110\ / 391221620557122767872884499712939762465351092558255771436334692714859871\ 228366530935127913035382518543189464308212314121128021908509625495386176\ 868778101803049916276266746381720828760134437178669300057885269501524016\ 552100350687975544936657384189267241705146891609634166379126454777861361\ 405439457561827845050020670567754768619374752203404540717400085535015403\ 9258811843020418208631363741500 and its differene from c is 0.25388294554850243000698976232466194565188860942533394691961605989075010988\ 929867796935667488211793787238404842291305571456086470343050650283544330\ 294713418196688551672535817016316864685409673664141520987369621716148980\ 215518810538643276081497792156821426377358333765384201175353865523708494\ 359532082128050547208716040268920900732849116926862177846622733973754394\ 874342075377977270536523977598995138304012660780753667771402663721003034\ 494293766942940481478392033123809890326124703118411215380658817055663230\ 150907719671334936278021884523436467092421774407125461143244025911310815\ 293544418761527418284909145441664106949127703303690557085152535704991041\ 875985994080358465163937667792794910960196818049906967189536618799320645\ 081210697786494564363432031439246283124697170615581236547988213613217092\ 804138035349974741889514132217667789626373261426348729110584205899615122\ 313313344696933239508783403503563725896758274613053632011661829899007712\ 965596016424852479759334904535964802502107417215464401552081898169353072\ 540083778594236818198900695526359637321206693694056548319921967729210365\ 163238422100828333786773609384313883838717083648618920823666791680261049\ 760552052852350393185129983859368618025777640687347918829369383473454934\ 121509856747760145794026628368297145263807427076009608701143836186664703\ 445646888809639003675891072176449619545328016971722014703370578930176117\ 195779767140277919406661021994469458243662264069190450941657747743450280\ 915168456113159388875060789310788309353649357670470303807513310718288716\ 959285988126174183351888896209452088082655345395695560193327819897067394\ 617378166201058101859461277699129636650726673580619514137970620501829775\ 720361391425082458602739141682671230026954151160776281605337123145960143\ 563637296282253554345304717919974633466626304374170943646171820902724244\ 559399004239553164551250750801172549019642661403874649480690099952524385\ 566396331992793808137669362628216663758424689849078269724617968102885995\ 524920157173586350957062973108239094288829054616610463879771317554570409\ 334612269251781760420370587032734571429017068321351252156663045141197679\ 453308312517899579685176225051352971879065193436980922261934102983897010\ 513616036880329085911231547993773299306042458988461885980850270733262293\ 784652108500204075671675248011707695127683421316576421139011593379848073\ 281345453810449404236057493851810895634237925425904802918903749682563186\ 570520220422540142068380422503303024107673458758514729717697320809234247\ 978140320146737262495150206760806472842832394160364214011579813861010132\ 829591295223222164940933959216965485662502019554502082534874476659006067\ 762357946726786927307274555872356263541647914412672335093315702291661414\ 778359773714656063923357670000416102791676323440480685838384934609340242\ 266121175245936955681405482605123778870536051028591352889376474195285486\ 090220338016902217261563359440718737412849834130644128112866441773048046\ 266910715393818991680024757176885031942227991035565564319564849187033056\ 431304793826180672344976030680248110647698975762466183392514185773697280\ 562486882998914192997162888110071560965429763110397094499465798098263355\ 321121771291091613233803036887949259610887227531115716480473696275964196\ 662954550255319382321574592895758027979975345323221492112923976711069302\ 944943500963126622115086084988632594929314343144940723164813797341934335\ 679161956647082513780784653400492160131697546183362618309067483672574154\ 569358270612134160484988319250516751876861823657725631147924331773892854\ 173686679846246024131375324522899378017399655434337450366245911247421393\ 511202284520594354683328237088177420700245427490282778422435503658255398\ 668308006051396053723779080290796013024919700255873660892080471693139922\ 052636677596069592786801550380516249946248908141456004157860227608848222\ 484597588976345890765689122269797958122514952114719641687538539504075156\ 894363088344678508675422978356090818558834468122558547811808489778307163\ 669675798353089260857347986351498815998211051448137605127820232099092964\ 392172900295117973082574368773765218750095798391323173084086383391600547\ 277447230696389393896539202630307172984388441600244554754904903574553214\ 014716374951326390833637190023045026640490330473510265672658876409113440\ 041189739427388963505601384585972521476532262484086347439939052271815552\ 418690480987215094176149509842304079987477479531590341657780272246374577\ 795564737115853034959673391824763177930818980251821304723444633474219831\ 259933084786444875520155522517039043006746239531105758010515408962318046\ 065872928630629109536672430280921955833347597092008837964451936544678105\ 061751241708356839462830182158660953048790161564231331456950366380469239\ 783605976453162468930648154754741628010157915758012545204683316840237167\ 983355896186283827267750349529923805769089250491113331294151654185756526\ 596922774832400138404559510848019385900984237587345204601853199593295778\ 205558894594274178204461720597812898368073655213935526410067106504555207\ 971754166050121755162731530121330171177620005472374834070825250566688360\ 006408738202710696484324315218982699199114964772760409642066763842503721\ 912512830160800382791752152031941529022282541152696191665277464035670032\ 713584310730531625796571556331828599491410653669232171149982829518902205\ 833124054904861536758447730179597984586127377887380793057234126798745531\ 099946831345482856346985751801211795980037590157986921356249879644999933\ 756140599606705058269281391648448164039755511455200298565987043716065731\ 178624856544946725258552717079283821848543760773669193606465342384191010\ 950883455424919839752111461425996441552167943517420855872812456937072840\ 999298004364024479238427290840719760345487833234176463014640078948884502\ 933676163199469089823775924434694827481617195698218855519983841518027054\ 061643097558872122967208070549197382023184967087786984791706677567145895\ 128389466869431666274532577376279354472032885088586490415118893446844174\ 229008013115458135027275782601768458523973132202199369720169881530905886\ 062616312082557680650100593916880361045604867371568996993234960049009014\ 323600968511404708498274291743409852548968323066426425817651286324376210\ 049076854478335211306922288177445553980205342210944685437577646383533974\ 930599384476749747972896940397977771562115879368496822569483316878525775\ 512889353772881859208195480076536287552241084282270076660560525777608192\ -727 1241075 10 The smallest empirical delta from, 100, to , 200, is 0.6204135602 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 16 16 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (528 + 128 17 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |17 (3 + 2 2 )| |------------------| | 1/2 | \-2176 + 528 17 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |17 (3 + 2 2 )| |------------------| | 1/2 | \-2176 + 528 17 / B(n) d(n) But , B1(n) = ---------, hence n 16 1/2 n B1(n), is of the order , (16 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 17 ln(528 + 128 17 ) + ln(-----------------) 1/2 -2176 + 528 17 where delta equals, ------------------------------------------- - 1 1/2 ln(528 + 128 17 ) + 1 - 4 ln(2) That in floating-point is, 0.614600724 It follows that an irrationality measure for c is 1/2 2 ln(33 + 8 17 ) -------------------- 1/2 ln(33 + 8 17 ) - 1 that equals, approximately 2.627072603 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 17, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 17 that happens to be equal to, 17 ln(2) + 34 ln(3) - 17 ln(17), alias, 0.971693035279126405 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(35 + 6 34 ) --------------------, that equals, 2.6157083647103328762 1/2 ln(35 + 6 34 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 17 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 17 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 34 | |-----------------| | 1/2| \-3468 + 595 34 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 595 (2 n - 1) E(n - 1) 289 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -578 + 595 c and in Maple format E(n) = 595*(2*n-1)/n*E(n-1)-289*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -578+595*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 595 (2 n - 1) B(n - 1) 289 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 595 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 595*(2*n-1)/n*B(n-1)-289*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 595 and 595 (2 n - 1) A(n - 1) 289 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -578 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 595*(2*n-1)/n*A(n-1)-289*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -578 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3602185255657678387724405134780872923889974711928648445154367562792080642425\ 884228454093854109325086649578792721560425465512579838929651399998395249\ 751314441738716558537511218342019104666838372646704224255247901616396620\ 423875261599829205236454886552868682693372858423516428667802702957690272\ 490304987374209230255515796580465857315724371809656925448371931229275311\ 258933923198559132555863233011645671309166859821928886179834470179784115\ / 2120931259 / 370712264560270577333652956251136341622368854278682175787\ / 498323986804310290514327372630978828693769743424805787369393763333667184\ 053635544806774527332823136412538577535396112688883353443380124514315890\ 181946368453710059860796908950369175878577495356752349469721600017520497\ 588227632164561647152094218177338536543302543081413034763727869393618387\ 432166340155710584933751700356404388245569356909795126526711255710780203\ 26277750769642078159581303360 and its differene from c is 0.15112111050023550370848318519748795879165310731553274901634783745853019693\ 519276698497889867775257699023519377310122800289685615628056400829137376\ 833938758169580620905918445320851411688778043896440939438602889430616622\ 854495646026356524678096525479760385465873836971451557796108306631432400\ 077474401780227981286531946464800860791447783632045146371926038107576975\ 045567099920200822954562613013961230743912068320717978248355987776010677\ 560289650140072061677724232532809891638298923205679876259188800613599920\ 143749898868263620369744038868366771219818971902984465338274193252373017\ 670858269668875107826946160396610969275974259515362615634846784051852691\ 148134956555148967843355110106004549028178784888657971187958111481834397\ 427565078906576230492985538108942700573630362131142244725549852932983793\ 147540220890156310637585661453744587325867306704510407106377218491298187\ 728598007028116120555568174542739939498575535471803108291484097865547134\ 226718725791759133014958350816000366996779817905945101183035226918987580\ 440618074246425126340321312112603483023885657032794151386106387884302128\ 144405851330328113783048386565159390274476417289385239603032918668496780\ 219100377563225863966936865137386355117502050128064919287074409987527747\ 364660079973073541272646485411729543465791245009731230984432096551029652\ 155483102123825235713874586472481225915450374115273272139966771369410877\ 112844110755045252471892404307050943045110294375824382390318068429572459\ 506413193246332446548377918298998946855420814280560228025185165139449491\ 881898507532171664815159216680654077167553462578515799876188540603999870\ 593214447717953011878577733068283248979000550439061268244794399929063552\ 995543704414954277602191523874183693121509294755377528514519881832119243\ 057992407032397179654461487810381435150925812687237736853843354750538673\ 150741639848500646927854864639250490598949924932258717696935050865776412\ 462751304186878780782267784092859813221769279810302171160407488439638162\ 913639827497034775265587392577315665020552975880838346046394346162633119\ 017376648391721590135533978691429731248815062785063421882143183822718618\ 200716104798168376409170238067944823595671548667088776062732227150631035\ 139319517756154960044807881174555554309484844683764859903096447774160423\ 054326236487790888317892752821960696634116674771567993172429316582570527\ 725455174556512984841748788102841697808365061702909837654599444358840910\ 482795606710997743277037599177844870471214233940639394653967551592673265\ 384386803978379531632772963675629086369045616222236631280371801246421346\ 343065796368188075210933617022579268005032968372261910807595075265299990\ 315692103652045078670665600106840578341307041394427721393434054147529995\ 300325360241156971194895204467855020724001658729784116507367796145870719\ 081076143775778152847480654464768862879685652645849377014408392359396725\ 955897134058246407368881114804088613511613277055319202170104679700878913\ 748905249167291226292731147418983323805634173441235165951269583083466746\ 479620512891748191266488357158019728409044052395498561463391917580697503\ 046033928762403081981383551400131056355896368062874743013910905159126081\ 757366668654991094359402474323564419887221572643552178635985922947404532\ 774754763412030717917801512438299659902235047502682532092757208940933255\ 045554875104689178771345157767119085640636696769195131352751138619748213\ 511812321247799556980462925599202298719985568627213242634017018658680405\ 636175158488568637324861871407196898521936248631857587056854388194036519\ 704070992768164063609905825476703179569885610235063568069732231448060248\ 897338310263845761495925042683793372388104830010056857520072442978703486\ 383238954494605725931099108400038217050058785850105576618385529461229397\ 729231162716430299167747433884086902481487102599719845868995699325729198\ 800521610153142027388369999598463140215177408450746926163083288348061095\ 098799474203884320298452503871365164302958332892341946965159323910082575\ 629226356451249254220964350901594491261118179253275773399398811242130299\ 694428374996605242270155014994658489282864412287147728646989820429770863\ 742995764041401303422570343361628838454214438276352343059882430703443512\ 072917595765256384636391642798964450476018455162862770153889883108104277\ 097656138847993852880290770142747075973957025666569999645037165281829073\ 541969053959096761719480946384961178972365317553097025589684455797035568\ 993019409090998341600987287155114661076085710586587797192431214268786182\ 620780499244616327641718745687747799883250533058003260056445381814255606\ 441626519952556744779764402268180402861425740535411529392172759146792013\ 582848963568522620150694737232339931285455312469043475585183943657381628\ 846470389079455538837758752927645304908681193867335307139332094932343008\ 021560939144176107650667801815721700624272568959255179690687833514924206\ 778713583723373136503731031125164649001335021282523957103544979262647006\ 515198177850861685515355656862558384559171758850624421043653562737694946\ 557107840526208114059689345848966709416102255623085491953341532313287269\ 473325080197276264630578516258836474022285180327533478360572565023834613\ 708713700193382469200930293825319593827350731619748734983399569764341250\ 555303096923586204635644328106697389234380185976359190248893171384060888\ 983067587221407137610400372154970280530396747537193039566639112373996432\ 668445751320100284005468666624128438870407473453326556431117861542539743\ 042283358539413015600180793718790308917044412761550513685038779099249432\ 107991220221525468364553900111882819937418993366908174206540431048192539\ 306034960269270867792032305881248882024161224423921538687675635347196011\ 448848453157333322330721685299695666379631497224199660465475264067136475\ 886349090148381613521691003336452813720335431650768723369442777462640856\ 319892368147741454787278357837134067046429285038362402946432909676795132\ 210408404507209096257548774394720172150879219272361482470463377735358908\ 718407218156821140761515782786024662557850929034880339242202007433719432\ 951915968746032401438842851391773329324008940338272244000038212134451645\ 807173727888826174489721451368816386155063035223226118728075503711660086\ 460322689379764021408360093756662857606765217856818023140192542868763884\ 044093461297082967107502773776408891991529076891591591932078405400417491\ 759798105427684888873904594258346746735988565310972182398552930305869 -737 10 The smallest empirical delta from, 100, to , 200, is 0.6341575336 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 17 17 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (595 + 102 34 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |34 (3 + 2 2 )| |------------------| | 1/2 | \-3468 + 595 34 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |34 (3 + 2 2 )| |------------------| | 1/2 | \-3468 + 595 34 / B(n) d(n) But , B1(n) = ---------, hence n 17 1/2 n B1(n), is of the order , (17 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 34 ln(595 + 102 34 ) + ln(-----------------) 1/2 -3468 + 595 34 where delta equals, ------------------------------------------- - 1 1/2 ln(595 + 102 34 ) + 1 - ln(17) That in floating-point is, 0.618923500 It follows that an irrationality measure for c is 1/2 2 ln(35 + 6 34 ) -------------------- 1/2 ln(35 + 6 34 ) - 1 that equals, approximately 2.615708565 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 18, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 18 that happens to be equal to, 18 ln(19) - 18 ln(2) - 36 ln(3), alias, 0.973209982864963820 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(37 + 6 38 ) --------------------, that equals, 2.6053484176619713734 1/2 ln(37 + 6 38 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 18 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 18 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 38 | |-----------------| | 1/2| \-4104 + 666 38 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 666 (2 n - 1) E(n - 1) 324 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -648 + 666 c and in Maple format E(n) = 666*(2*n-1)/n*E(n-1)-324*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -648+666*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 666 (2 n - 1) B(n - 1) 324 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 666 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 666*(2*n-1)/n*B(n-1)-324*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 666 and 666 (2 n - 1) A(n - 1) 324 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -648 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 666*(2*n-1)/n*A(n-1)-324*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -648 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 6717868277720208714696877668329987867066030278247577158039292209698300512423\ 582215513231094432443933390862882465081141303636734049777641610382670878\ 686575932541038983393621292926776673871361169565724137475699523369458857\ 906207723502389189919136205295298018024469627856619101223083125725962923\ 996123265414115179387555532479023050628963239879881439619935662161064576\ 381371232180145786567368271980944859391922748401363834660269242090386291\ / 019474906470870885269 / 6902794254066273464406799729214307417344653783\ / 420854496882412951463447470099609962151771636768495107484716983301310883\ 571680747006398705358977105351052762123713528026804112603609414718201801\ 173013603659505187323925892750458110274807576163860515560304983392503255\ 996190276894485435306718055063593860137209875376639647860004175156166422\ 064320352530419871364119056474713245656823860947176224859255446722524210\ 594134042517865382200498001925380469656925290196000 and its differene from c is 0.33327486134643998359966726157342874776028468932962793936430446392506194570\ 614064172671268834317105736902733146079044890573700403674770588647222351\ 605983368864832223464397724772972626326492733616092414260949093492885849\ 188225613648510645395160598418866245987696995820593804937774476193133101\ 349477235276291805814114203122633673216219021106704914553303163709707736\ 568336366398190196936912886633774091834840757465810815433596355593994281\ 460966011302359083467839095679773018670282680846145797422341157570196055\ 745295204010511089960283060940157436594966338031501163027563064106060284\ 827569851514878283983683531353971665115126885282778416869527533602092398\ 740035913051623539918289738204332578472474351835277627804019957782948828\ 726979857174364654184752657864830363953731507720416300669593550211295442\ 914427614330200972866970509222836686835633250268364247344620056950150923\ 368007980848118459502024444634901529031628172177687617788429978492910481\ 373107524107442755775463436892184357862315179495437683969251426468488936\ 350114919474953251581530746953013614016152101379512361436533155021349720\ 442546331658147763035419493841686753001393204137390776003422731934748718\ 024131272933535949789675903938328312664875018784416750625113723786032948\ 223639194474989334306484557596450027098280671335525678713155775255678567\ 891123856130542952837010006527046996653640481155321285950953507959265635\ 855365944211784279907508004842876897952191953938624177792040628636334027\ 022551397439787483926631187125626390085248991942244445863873442219040525\ 491050583451196070420966098183135822057567603847748907799382671942081477\ 368061571257790770151433221935281811716810461871653602237405051585026748\ 291003485331288884084863117054714557909000579237410864215201721938724842\ 216614423694443951565805580709723590250651137717164144699889727403413010\ 897271379480522946602150016669197689185300067693697907554451814265872990\ 131418784825424243049609486994300905252740303579813318145429079044432239\ 725242070076808142777913118869687949153430671537691046969829987827927351\ 945901648666047032155605568603803101022700320796093127732797923905539507\ 047050018997976209797385579740377809075068044922730737376384361916791026\ 057080820217241555099143886093355649169764660885716134091029629673461932\ 156413244892716869215187524406551597498323851754082318568627914481776549\ 573461740926867312032834328783983308442781136159974530465961682617438871\ 752089778806002528675918822595682701575426596131889613555691413773297863\ 677971179961553287810280446795793155804101687805941217862442931085369935\ 876834336733387822073378180356092157390649777511607112849851202730180460\ 639244646382654793834405292761895579885678412682860805405309175911740726\ 413077321471595737347867570032324942896992834048674823418620977971979527\ 450016752526080008923228622365219787593201254257760693768037028313731454\ 316687968474745829948364325496730564637142390836333543325406168372837653\ 806458895484813771957080895942629085181790648313130533163437352835494815\ 956149389922106410824699590068434069534389458899916170638935083816036193\ 490721176068071310972329408990006402558275804497111588847791818032310014\ 224649549599027380240022723612880786456719788773326221586310735350788634\ 078664361457823378952938685887961495629448023373533232233935327742244882\ 388917362562169825521103909320912968608611384219200147674427005715755310\ 845510029334061770965847777197064894086870863566308977429683784081727380\ 300917237191895337369936725366362904777035266402251099484928513943822274\ 384236336412330739362830175567525676989445804014529629844594450621837997\ 035835167993851349771747237454077113832397899330818637987534930546297016\ 017588201569083920819862097101020597567438233992574295726080206923746257\ 187183410823316969156132096574850997629009718830031502111535523377661848\ 203147523982749944335906945203916979378691268113944627569598331065103436\ 975267869106247035035779528267692017274375795325193222596803782195974750\ 000764133779598566374505855372490843381600404218813525450563039278401228\ 574058155683104423576052608202555279523036337934601118425877527699731233\ 432093412950986312674261112685613347149985313678721340187069425321069913\ 041063055907168392632465153478780224053821438182839503144178317754789342\ 518262021318165867766408224333746778947972014106278602067765268470660136\ 461815264219319900331406534686154371639170320869311801449206228119945041\ 681712844967731288123479948465347058258651571938306905773720269197251686\ 899480011175362436754193741580342534033653287351961553195809423526535052\ 800505338480319853695218612376622939329526352238608527965512649493397817\ 342174336334250664582979580964524738681203098776244414312306561671741025\ 449598141656372916944806293398622532277337615811183467292461544802529699\ 119544546576625225675399937948949223851459980715478304051476050804459963\ 225337913975430628672467661895753716562656044030006957174246988877588348\ 117433373372593268681358654087034363990822850614812006863562700453337406\ 199937167530441099830967807406852116367736002894347109593721641109181304\ 532958707377757233112850507882299828912305179926875690516937370883236984\ 327082857671984239052223033122036323468596993272989378640806484152571779\ 444177402693833561338680531684571470332137323320761628821599220351601477\ 709799926911884961853198887176481558779912607287985903648623444989292456\ 074464854423630345498313046934300501581682006713140688929911783560722693\ 223658896173108125025149216513636738436081960776581713936996940944685085\ 669432707537982985292988596209448212464554807758128938437097116588960561\ 406412942500007848299885606243061039900789868166871976018358006748231078\ 235944433000817142155468450988735603193319639303606768891944670135325236\ 824056188599527073624206834747354046709724720645949042043840463045187531\ 613721853692253012506136057098686221213434553275233256578689109167791058\ 793991177278838002919560506786829886077148814285797677405756801340021048\ 814315538958694853605816297839659837491359225809167300891975952274860687\ 520657777557828032176986704795976994063467073384437080555706068670434432\ 175380170981857497089763829516077477852265403718419444207446806337023172\ 950459097736450338328819215244074692967589893823252657276460574643943141\ 539893144109462502076458054546140476483053370334532273483310008642309228\ -747 94602944547758157243758732351540312758823317665247800242318 10 The smallest empirical delta from, 100, to , 200, is 0.6305381179 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 18 18 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (666 + 108 38 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |38 (3 + 2 2 )| |------------------| | 1/2 | \-4104 + 666 38 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |38 (3 + 2 2 )| |------------------| | 1/2 | \-4104 + 666 38 / B(n) d(n) But , B1(n) = ---------, hence n 18 1/2 n B1(n), is of the order , (18 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 38 ln(666 + 108 38 ) + ln(-----------------) 1/2 -4104 + 666 38 where delta equals, ------------------------------------------- - 1 1/2 ln(666 + 108 38 ) + 1 - ln(18) That in floating-point is, 0.622917732 It follows that an irrationality measure for c is 1/2 2 ln(37 + 6 38 ) -------------------- 1/2 ln(37 + 6 38 ) - 1 that equals, approximately 2.605348425 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 19, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 19 that happens to be equal to, 38 ln(2) + 19 ln(5) - 19 ln(19), alias, 0.974572593363460135 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(39 + 4 95 ) --------------------, that equals, 2.5958508958571879623 1/2 ln(39 + 4 95 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 19 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 19 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 95 | |-----------------| | 1/2| \-7220 + 741 95 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 741 (2 n - 1) E(n - 1) 361 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -722 + 741 c and in Maple format E(n) = 741*(2*n-1)/n*E(n-1)-361*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -722+741*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 741 (2 n - 1) B(n - 1) 361 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 741 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 741*(2*n-1)/n*B(n-1)-361*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 741 and 741 (2 n - 1) A(n - 1) 361 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -722 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 741*(2*n-1)/n*A(n-1)-361*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -722 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 8316204247668442200323715033248821078384856044347335968082999462439218649048\ 801776303480836333102006463009655711203657048550808159039683237761057504\ 300336163177803545257253876719850855001045041914860376785843889997306552\ 580862590738944876415278372440192117974480697110714372014567717940531897\ 432045921245953594266113993450716621119168333245368758754958947603527202\ 677626688059551899846793393773492387015149806352383479710604926027025924\ / 18493073571150611810137353 / 85331809085328660730099321798132346911736\ / 127698478105137095181183291188272684423839714018740848049475891449158420\ 326933070200967168703267982432292305966599927324009932052695866789068528\ 429471176305400187575069372097812566277020206331427413546753793204929192\ 705454339572156279304624117972092353787915803555008443680447590561265665\ 860354097493304091275868832017177629309147699198760232986170104803629045\ 5411030627194745804145579481730041908742753984605464949928000 and its differene from c is 0.23718529596001675776492496381436806799025704737417181021430904228421264861\ 272000593779461568742499569970261676602808786507978801347502260412771594\ 409073713945816904244536986100388971123456984383606099148542926145832799\ 856008969220195719921765105816974316021727828755431730925613611786352436\ 216658785545175816054553541624434518203533082115071932250364411894971414\ 658158101165514773821249899070289029366056815661469916335244293717195120\ 044636809311639994751599611031123860888687667362204268622789902727893615\ 105234845147942558817635521637635321036493191127147414541974968486922175\ 950964110590076168754504772603613697536522081455721967123292892195305076\ 180496300324577246314495132028474529641416005070282919423514480128621874\ 425854238861994290375976684552046865068055936083768554612752378669613765\ 534862467685447720136654686491354229111291972382250791720508409578520326\ 824017021628308944572197587344268369788744330998145237899705125633677329\ 323511463226548247397604500012654128063810302822634801200790719432206086\ 735164991475659805690229116602321503940741766456145571664093397190181343\ 349832114567437050868428361167254273296525610130857105215571564311078650\ 603406185752448313270249296788208688282741255167309376167547388611087367\ 019823410042040024823390505674544016304917921776211252600847503220996673\ 885760283134520539564674620371068187710521668959615587494519576076871047\ 080435809660295157523376246000594579066418684020181112073634690445290916\ 021368469966193446819187458248694298006170365063956379672187134876817765\ 839930350181401096978340796657571621518999985301989122880138167732646884\ 006955578711913132911068318033499894111742461955007246606281227616552302\ 259802955297643141352208267194624183046550702513726493988139941078485925\ 787404342105921175515127742543055604239670504137644241604725170374373150\ 631244384395774108700558924629827507248539818135537423491760562040319186\ 685197289966569658334438829053618936111162185289819110221759805016450919\ 378997566378069440203222314811536673392428662837917884244430484834534826\ 244647636904790413425380431112278051281876717315124222562360948627379556\ 012527670583528774174770129422609917117127854919257847429491678900331577\ 962578157768811621821137201355182257322972450305888215599962003997481482\ 551863383485526064415736613328812288834520670612452381829684039955220381\ 335148113789737792930045167186862365067593215446871533769033892210499235\ 609954350511921599445093522897428915907541729142034978897476787098906804\ 684325495452112597159321362703606305652238885837524682313679756387209185\ 580855619967598949972238543898496643436530411770067947264825523226766977\ 295093848366671300396740731001511659105032320689187102338107127920185546\ 170400366830358066916624461507566024040058406953406286658514977122333956\ 332403458516320121882647544567943735672931720823273283280156709364733647\ 458569966445772142512061556927251849926697259907036239252284182902901823\ 429272348722035279933856999168871858766744574513011624340812562724317559\ 939080997181468846983844246897919782028192543301781404339546942278933840\ 931590735024981011608956563846296600379184117265368113400524448504018071\ 483247055636198369726830096847388005444325085580495309947387251546995538\ 168473344805090468534242030413205267646609346630409394014380788820472720\ 247920834356048381823144693117037038470815824499984302643782919383555523\ 574136052914365499671414279742074980941231718918281433545199446630996686\ 416587157491828172455220750771533850915007184803065483278440178155597148\ 167930026055845243915593222191349053929998846473925857835157957699790000\ 372791864648696594012978685005200110290150683450531212766190692203523916\ 394072111546032438937909755400263349542831536314021005515319883525152975\ 574762079629466638959189958079126333643844346314894625440169824220707210\ 048956158670845458574853150096256386429717532210435857994579063088701004\ 049046403216486997299125111141191165146784968696083867585839838215967182\ 791138827017106066364183608358277475120787907645912915920488280271368632\ 165886685750533287155558846718260468217643852037890997683720059493746872\ 494120592990157294806933509647969011685522948128286119289676431515740142\ 009772316497301369001590446901771098206245454715264652028634974185481484\ 305116492734621432880818388885395032345282789346439131027378119915924875\ 416868532140078270612427886644065634920613457754681210725058728688089713\ 237898417665464107898689017151387532282368030486313176999767409973765186\ 527227355087459884012343760211164394023064137927186518370226772021570174\ 417785207658588654875849520944517893346642877539471191933096943487299893\ 906574760267339751028808313881729910831535728441627354724754898457814379\ 253942874486875160670404949706401865940865640400157807189229536813199543\ 475622862650938356998436454283607859834199692658245756357473312260756783\ 451619195067207689155803486414132311388682985152072499906020173691893960\ 604487921017536618133941057119148444258784893407086026550652193138752699\ 974822008895193512550704369991427953884536444128653347995369490257325355\ 016427456842017638943409653740677729440250444474089660548627533819628639\ 774097601707720121441656975918416330833720793236805788574711031381588511\ 512187732901723081801318361017131078077627587132512946555991631945897502\ 100923567701713883479165197907115727923816256918860317968707255177903729\ 444837076229215088055493676116341931881477591718117217079533815294360767\ 478793276681050961563092623643680331806956745450806658664921147774044692\ 727415040933836564236041181045750562900783162423960821530717592028878070\ 225961519276173637143000772206588214851292558893346318326601308009988259\ 499549777149838192032017785326540306995949571769602285992802068501474417\ 261615642963312973267425320442982603394787500943440450464866488183231408\ 412790455970942298844588657225643311279231464763853561872557792330782003\ 162596337765562538539784008983721203008885198857815258741594404955375242\ 115244834089463339581299618123551171440856522729846474438722933053265075\ 996991548888689629522200010883764535556821771816610392082818427282681655\ 536251358541976064930584616060635356780090533136283810874634731438385984\ 332007777625421929940258418447396958055781325948119389268870048341013921\ 855966465916048164885204784147085673160796564840111496136880120056303494\ -756 21395235556602769371312848859680346114813361282430 10 The smallest empirical delta from, 100, to , 200, is 0.6325629547 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 19 19 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (741 + 76 95 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |95 (3 + 2 2 )| |------------------| | 1/2 | \-7220 + 741 95 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |95 (3 + 2 2 )| |------------------| | 1/2 | \-7220 + 741 95 / B(n) d(n) But , B1(n) = ---------, hence n 19 1/2 n B1(n), is of the order , (19 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 95 ln(741 + 76 95 ) + ln(-----------------) 1/2 -7220 + 741 95 where delta equals, ------------------------------------------ - 1 1/2 ln(741 + 76 95 ) + 1 - ln(19) That in floating-point is, 0.626624918 It follows that an irrationality measure for c is 1/2 2 ln(39 + 4 95 ) -------------------- 1/2 ln(39 + 4 95 ) - 1 that equals, approximately 2.595850997 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 20, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 20 that happens to be equal to, 20 ln(3) + 20 ln(7) - 40 ln(2) - 20 ln(5), alias, 0.975803283388640061 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(41 + 4 105 ) ---------------------, that equals, 2.5871007231655744608 1/2 ln(41 + 4 105 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 20 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 20 ), it is readily seen that C E(n) <= --------------------- / 1/2 \n | 105 | |------------------| | 1/2| \-8400 + 820 105 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 820 (2 n - 1) E(n - 1) 400 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -800 + 820 c and in Maple format E(n) = 820*(2*n-1)/n*E(n-1)-400*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -800+820*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 820 (2 n - 1) B(n - 1) 400 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 820 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 820*(2*n-1)/n*B(n-1)-400*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 820 and 820 (2 n - 1) A(n - 1) 400 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -800 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 820*(2*n-1)/n*A(n-1)-400*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -800 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2731201773728995055374880548945491676172135324598906887334165057466009035899\ 495368012633408289647329414988086031338664678897475747325071859984149929\ 541886039447561961747631707191705104302596063352507308621262536172085542\ 184277836756637121054421863627748544873487982028513991565798727873292001\ 945750155230351533243850128945805805353399627533939295730694362440927506\ 203054105947709798175083415235251197839379652987074754473522399343731256\ / 79555375305674281768347674733 / 27989266076708004342454554122620979958\ / 272508151214534227974490168026019238066567447280868109631784018514627637\ 587892731257839667303129649151117027052451883548511379193466022015652744\ 982286783765715440224198300473867886667014646673132915041499808407620324\ 546006237712263896215450861138989896111768270775427252502117187346023894\ 907886626964877596118477748572819107340196635576952131545833999764607266\ 8967468101897031170336666290116796311353150356193788843918073032400 and its differene from c is 0.48441684350742044635617625232837323808842141160304985155529831195841096011\ 500669180288939986142429874411727583132637559172696131987893794527176039\ 861350141497861673393762174743524713036835097139612047021883509349495009\ 190765678062123416476082376963713350077627416524711736288231631224379432\ 051606903684657967635819089538445901510024733808708362915575632153643265\ 538687130023829116921495662264280690812037712421236961052726456221833960\ 353281568537358647092289582363752236506530150730398424756391248087204571\ 339190590061359364136229006231713466236133930979686267444837611650653347\ 657088072615497634764178061650576735871715284635300238807242984641037017\ 057989950677582574146950222429899464074415068289014462441368432977591516\ 094660755130309379400327834757361995261561407755066463179280541779845578\ 523512440857682803200063664155433644031981266014891849266175985411295193\ 536883722180658126284004810325014144493872432638996030446575382174206583\ 900915402111071361860247221304309746462713649935139201112135201174595735\ 963560519084350872504078574439876417690571840320110558538062491141174870\ 703752062344444926193591610260972341434747145245800837801499531794989176\ 651536678731208919917403702698665841865519628914180781797958804836661334\ 297645689554300236319313997006531841510796542935525666467519694219460497\ 691666847468254420443501398953301907578387264502589345760692711639125951\ 524424598631671129780716366776732792105201706315716218380993850292739202\ 174074586986929571770157362342503321677424362832693662402343230025281760\ 043956984254547932992420498267701718470389731349078344871342598475638493\ 625656695839672493651017809976185027120811726643356400268457463733749889\ 504062319475302730255872172420849072720708945001558117626218057216228386\ 923267336544741966031006339062630830162389899961977459314585995989840033\ 213667467835107030367740747116875397860394948589689833400490929660813399\ 001008667217201879364674862774202806751481316463260325809279458624884658\ 333562051586277704129005078214544209407420280062198705830948986749935930\ 255063596486865376653125880028871286555508989743717160834143355175762065\ 861976047620206172657247761232737541243138943160804764554269806295330097\ 472614105825509509442309618283818476090497117591478922130990469411780238\ 092364298612824907991021558407990206247351886994207310296480042144949879\ 695374243784088774827683647002281585699219871784446507731615472111547022\ 101306188393754182796975428661321880249212284585806076698104811675628787\ 380686363398665102150494974446352409448125951185960514936993202451009678\ 669515624818707125992241854045713779059745427907673953743397231734149989\ 238338969058305177551267887162185694519683836668251426683007470586362249\ 120665315843416897695380143962657485990529209793758484282764289782443212\ 840328579752685064055191245427671684011787720497317357595634782727045076\ 671211761387498736046128434035369322322716672682597111694663013936631850\ 725189525289110146636876056060186961727812115878218616568390098218628079\ 299296279374950060127072702183030391430997531170340318741786054912365226\ 235214308844106421888985734224315455576837956323540871237659616990047023\ 278628713012945801388049587198998177935985590100455821220848008315868356\ 663501604542658015275848692337649523673365074467279294822358474587622959\ 923718155285148572138916471167966287178128446821716869704375175032192602\ 159293995595774167357114033995216712645089280624561268041432941587052544\ 196346344986932459534869381845095256444229752681965819568783878552140603\ 510007327321648812967377535909673519171518148945653754023721890506367565\ 628379312051236769050609130727316580510065699232262827785061315411212235\ 882988854749237591453434898272331659119455788719224305903513247201517498\ 468579439662534006707473040237660268185508968177987038564392301821617142\ 996822756493509654463419572687942410624837580308133809745608320122707982\ 986791964632384960280496507227225625903031855318876474628986453968281504\ 006034651029883596152448589134006810428412850983388615104823764544922906\ 956180495883284894552316177785275416942476647476164919101016950563805687\ 713635209126122660724927109280462989764695471599995669899675962728904378\ 574991726245400357753012042288487700542858781495879351737929796161931373\ 204741015958438024466974931765182858837098638801464379842982205699903001\ 304174410317465300233083843957397348413669700928694368528197525559760452\ 775151674485020408175888976062120067509002439170542912061326036140546885\ 833620847023805860665959851059394764117873046376249939160343254188694765\ 484072150981854419614585424470008114062338215516740466407125441380772234\ 037763541987363350522207150299553335884608171703462071117850137981705915\ 385696437539432946435111688788552527565106444049025965139288698297862245\ 084224175342088729603770734177122529879195118393702773186402176736600978\ 394287223715398028534005648767455877030652194794503816512695569766006771\ 217697089189517060269893939501766346883925624575787567586157734987050002\ 683598418912090407500913753444062873584017468540807917747690880328711973\ 446077572678440111002358580132857788336829055694263247607443217212941721\ 928877789057196237203805851629359862184594222153290231401959535133503633\ 076411801950965465703362004193344692301014640867088291764606686118732405\ 263242963815759598764260722807165653846121987036062561769324173110796183\ 600715578008216465322581255534550610482272541905481622004331608562612912\ 443990616434163092714637049130466658326781501747621889327219956863952207\ 586851984019946127864269161552980802251308502231547340466284696639840220\ 645798491134460470322558587106029101282294381297415690676102457805453662\ 812347951067013817502581396126652960121473156976929850740858134355169868\ 258588882229682357236417413980898887039039603643920316408485935567668977\ 216542080247512316438943224720659255249498278348831602999009933344029874\ 716427728847460605262919998176206245483641723096997264032282074271161499\ 893667477216582941586155212674818550915316428215945358379197693058799175\ 228212766963386149392433991892631155878032482571786426881989295326424381\ 650675602122899985904861557414980210283193611883591469839613933065845454\ 908372708688075070385463487800775227448022523638555466589213227802085363\ 426299933297995942254643442572882145596774474447096809393834720840968894\ -765 76072599073144474513447725741789953065140 10 The smallest empirical delta from, 100, to , 200, is 0.6418026632 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 20 20 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (820 + 80 105 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |105 (3 + 2 2 )| |-------------------| | 1/2 | \-8400 + 820 105 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |105 (3 + 2 2 )| |-------------------| | 1/2 | \-8400 + 820 105 / B(n) d(n) But , B1(n) = ---------, hence n 20 1/2 n B1(n), is of the order , (20 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 105 ln(820 + 80 105 ) + ln(------------------) 1/2 -8400 + 820 105 where delta equals, -------------------------------------------- - 1 1/2 ln(820 + 80 105 ) + 1 - ln(20) That in floating-point is, 0.630079518 It follows that an irrationality measure for c is 1/2 2 ln(41 + 4 105 ) --------------------- 1/2 ln(41 + 4 105 ) - 1 that equals, approximately 2.587101265 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 21, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 21 that happens to be equal to, 21 ln(2) + 21 ln(11) - 21 ln(3) - 21 ln(7), alias, 0.976920328332749998 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(43 + 2 462 ) ---------------------, that equals, 2.5790032473139556408 1/2 ln(43 + 2 462 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 21 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 21 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 462 | |- --------------------------------| | 1/2 1/2 | \ 21 (-21 + 462 ) (-22 + 462 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 903 (2 n - 1) E(n - 1) 441 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -882 + 903 c and in Maple format E(n) = 903*(2*n-1)/n*E(n-1)-441*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -882+903*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 903 (2 n - 1) B(n - 1) 441 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 903 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 903*(2*n-1)/n*B(n-1)-441*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 903 and 903 (2 n - 1) A(n - 1) 441 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -882 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 903*(2*n-1)/n*A(n-1)-441*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -882 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2851509139206039500815323396012530635080305908320633472145656537416140965471\ 248967336152827008179278150733277824396821926622226787619086349999088421\ 240471924376060903849883546574841690994008041655297162190442180822019061\ 090478709440764609971828624175469735579674223042079612780789959937897581\ 504526561531556013916733911650220891835348647191091470651445443178298271\ 521182557461378431496291480934857779406411401500600646072493670888801832\ / 244799255713975149662152385837301 / 2918875835118033879984462427416369\ / 210916385631927234796613140789715205020779994902717324853339426894916644\ 668725606738641963483169367902419211289229141997051027756256914696498097\ 773754901362098946259752736755545866909824795178077511139482143551333732\ 996535292106098167132101319427119001460031564427803659829090111404387555\ 936292797565925928400134615978913724066762352556239991207713530902657428\ 759705778758456908318630945624057434684188947505361133114507103480371128\ 000 and its differene from c is 0.25676163057442517954448425162972735198039566283842459210837059441167222736\ 965125114288418373090118550568261065775065235466699159377461400625007899\ 368562355928105865697234040706878119192583139083441497556815009811647042\ 524010092494079397041746350232970047519816752463458780354541946352288576\ 227886377852104052686117270451699046700012860429010128964263789951581132\ 859075917161076045817453219953530608272380200602073266719600557397820710\ 666937581118782161233393293649525611602454785573197552177889831370653077\ 712620902211674236128550698453680133027807216225905994786075980131046992\ 903041330478998747703926473935113927003056179113509303984672375618745013\ 137211490895890234809753661199129512809621027157123214684158875182401949\ 027000429439443029281002496667347774192911970819047540359338918284874767\ 931406354724628271386388100601080249787682973108789267674675950027763801\ 502404714906660473729715576202621779199693389062617854417964897392532599\ 385202921297468607358649016949801615227417785358986825856362669348059061\ 181231463733969527594272043334311945908473027727387657842900910716796248\ 169177027814035761725719201798871548986297296251470008326870412137143719\ 553277682473419581709251117974613998719020891443388748569423682303888724\ 862985314904843554838738720724893052794293394973108524683822008764069713\ 074357319724386695251771420677328377917072765333041605635777252483893492\ 032701938343783931181036984205674906222013120934481725872255178889196056\ 088217640991107808096881915276706340436605283879252520910956100677536546\ 775752644719583019208987000269121491540047079735322078231890369673849681\ 790297161593076218788755406589747214857203146863641477050631212695820276\ 173909001854459021087396185491928997126677677713479675689173261989306277\ 529209644126474740461506113841939196130295533712602837964949743949421501\ 494122019615435673723278869854606715539641600078389631159666667589878990\ 582961716470393512684146487192480376709971090343031198091053689846803327\ 983232485216550782329388882166737165379481635188437513956744579198952219\ 833105001221762974049736796051936765210809003371357992208729172903301266\ 907543048688725452868643784549189335661201118695103674814927304408382692\ 129384122179050501269687826219582233668623081717280498484111610752288095\ 228109493522062070348333703367949953127755586971771406420547457001351488\ 047128234750804416913105769860339033740102266364117143209788122842195576\ 064529072190047950156126260863399195721645642232214174901084536312704329\ 817149690257034909210412685138911260914194750094422144479588724051853425\ 330952617450592470596127808290801900770007759012997899311580517349982526\ 362908174095975963204455908023079232913193996142206213768216310167206609\ 437860777193609159823341162810229725339517264430231006778824178601828533\ 537396086369134357400216083671729887083300436721674577546835087136527900\ 578952630552714917527299833733845265286368298563510094944928195462898504\ 130576302685882369920735868556712292326765609642203049416247203046122518\ 631053839598594651956011775782002758682390859167979128797976094809252069\ 281140034760918318730769642640847951137765851971764176324826180117925697\ 990944375860922194665002904845529204006037230490366256182090864924180631\ 382929428329161007995144559727417097015353966837269910758525379587292731\ 357971045769158359790673353493851769531615901133380070816399542847623320\ 998471201748945096432591765128982803634233102949517218228433905669429687\ 665136211388203932578627183158385042970629856543225626357114967877422538\ 421651738833537224525487548467485247686179862429312396787488151858606321\ 614573017931685965071113202691808339440249366298158602401466075114776540\ 173904208617993864579340221642900687252801705604358796310378969087660582\ 315495550511755835749546445711285299201166775046837100192328898489697742\ 611386985762920878449229373057771864428403175824984585530953956815821535\ 997075658999958459201416290304341601489374425828123045237725103716273416\ 156652188217449629545846176697716087614628198904029339350235355774439741\ 186139085402953488995261351188017896212846408599634102612505667847294310\ 505721936885863037136685053395789869389907113585994593311071664120389247\ 512386248709783171593341112404806918432167211549685946908282891772261063\ 883049365405978927947010572225160806192758722118539637130449937306766873\ 559115958445863630756679149743681206156754797386879194767452164759940066\ 543281524871848651691468083348389886616794763434785381779953156136510552\ 709132247432500701666753500200005330306715915294158560447572160255625095\ 870259053953760566295817353993344986153556305931211558980246042656288023\ 841001406392317680448243150818656742710692637313849368080889267991122649\ 534254396683413313962203191755201120105594555277899306431706300432256355\ 829579173609977626467518319485492890436445647267998471009313696794431501\ 039962929664951768798251793260407288773111169152142374928658452508844057\ 488720043337245692183403172162876452950965045613974397768852513531322493\ 106260924242679312763116285716459275570476378983286291522295502119806662\ 395621151404787067341383980207442186461880436828333821126981274321907549\ 925433863673242748097920478004747402654542820555897421316631516098822892\ 371800961518164495574648029458371772604678277564599412705643028905111426\ 693431522454954082477549952182531496341691263359524441602445333141716180\ 983718982930011114153887500896004781219226751839258389494706544854127666\ 411520857997310904043049600502117318976893712096948282672058413721421885\ 163412989210083001472347037513433131194100885192141569215886221201899680\ 161198852365024488904328772998818438173695099633106775414452478250183925\ 111685384353845597008428191397918002001984323547605138770213708387359709\ 407819853351667954792907616881226805842001693606826545965063445646280933\ 997635527008633666855890775519107771305573631271592249274616945294142338\ 457521121292899336750532484390124570037591032587817209303325568799414867\ 761535682693522079533664150239263398276865720841694164302036049898565811\ 641447073337763284838908736901042411211815515301744254484662716414757688\ 213794083065964204730580958256785870087064473335516927760926116959336642\ 573003452246643751686678055992512140772277675679461716284051208010067710\ 229678654894098431025824501676331323229530238475150811843402421813189248\ -773 564745434210844587331996043124168 10 The smallest empirical delta from, 100, to , 200, is 0.6433166189 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 21 21 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (903 + 42 462 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 462 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 21 (-21 + 462 ) (-22 + 462 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 462 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 21 (-21 + 462 ) (-22 + 462 )/ B(n) d(n) But , B1(n) = ---------, hence n 21 1/2 n B1(n), is of the order , (21 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 462 ln(903 + 42 462 ) + ln(- --------------------------------) 1/2 1/2 21 (-21 + 462 ) (-22 + 462 ) ------------------------------------------------------------ - 1 1/2 ln(903 + 42 462 ) + 1 - ln(21) That in floating-point is, 0.633310921 It follows that an irrationality measure for c is 1/2 2 ln(43 + 2 462 ) --------------------- 1/2 ln(43 + 2 462 ) - 1 that equals, approximately 2.579003246 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 22, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 22 that happens to be equal to, 22 ln(23) - 22 ln(2) - 22 ln(11), alias, 0.977938776558344421 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(45 + 2 506 ) ---------------------, that equals, 2.5714798101987115606 1/2 ln(45 + 2 506 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 22 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 22 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 506 | |- --------------------------------| | 1/2 1/2 | \ 22 (-22 + 506 ) (-23 + 506 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 990 (2 n - 1) E(n - 1) 484 (n - 1) E(n - 2) E(n) = ---------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -968 + 990 c and in Maple format E(n) = 990*(2*n-1)/n*E(n-1)-484*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -968+990*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 990 (2 n - 1) B(n - 1) 484 (n - 1) B(n - 2) B(n) = ---------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 990 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 990*(2*n-1)/n*B(n-1)-484*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 990 and 990 (2 n - 1) A(n - 1) 484 (n - 1) A(n - 2) A(n) = ---------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -968 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 990*(2*n-1)/n*A(n-1)-484*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -968 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 6773184204261114518866980717907541681631290996981659970142402453688274100622\ 532179125079611802098320670040565824814290559275802427057348467364016283\ 003789326124970026608945001613522570948193905783970747803808925672315352\ 727452409862890927093411997211913165090476368150127776690090005584699471\ 628645386497347752244631967378512243315718226513850973150700653013921535\ 760696310789724035325294020339086365964773517764134255773067453788650071\ / 2148761934179557782242664483668628013 / 692597979200492643172450603504\ / 209022979617606729871489455827351629043922131307386192621467316190976873\ 780182039768406915682498653314695331027685771355459337879679755392131897\ 517628293970357727621580814017809014910425622559143451946320167974746591\ 676593582750022130749357709465244982305364306027536712626805369461339849\ 300615927552509323584307988546902379159538797078658653069537605434771044\ 442449900108737269542731201860813882970818731811314102473869291269024638\ 03708094560 and its differene from c is 0.32383132498493869997804507813907505478430179865205993246445423584527182237\ 868156827689353401770628712122271575337999719536111232891668089808863188\ 856243813700184283470999547610816449999872021911625792922094269493683514\ 608813371064907902438681179625299772718675353760679948979337217029844915\ 641002828124891762966546268557852822439471296633452772687951143887443047\ 752646427534027638079508794853300370317114470770090798366400848855030725\ 434059891164537339301714129364151259841745894317982576513415244115892674\ 034567211008741598929376593998679691187600726129353247296299672910393934\ 922651639508594335780836295006419444203015835816827733169144742987789591\ 913929867368017614218851766660581226303943856477711942696232133987975362\ 511596967522001637373272230342836607392008211223608973552243219186241147\ 053089405827881725601027655350903380026123838621380345462691912622571079\ 781651221750733494260171043784805422755585496640302217457376620346160238\ 577051739073503701573382033933974842407850928581494889852681415405925246\ 143626295334900586357831185684334485241766973358512801727529867778247720\ 070603249395122249999852540560835858103095053958511460806415676475886261\ 737597348792232788184821682873129916949378870537040270133201505466270150\ 064767682399159681551770143470155230169730459485807901354106302491544431\ 848709956233273943778848174356568635219047698012609612032531344486181830\ 469132723176128093281352571704193979885312200367166213864969443544993701\ 879654698257097373079422439028131666193947317750733670872702484018893577\ 067891499213952740158789565730329475358954904127289036841829331435052851\ 563799545352111999533903355166104584225900438486845134075483562142774435\ 085700688819556328276700621273157328656064367137233982283304690596610803\ 254590047600229018842092183563810877924505572604483064225855153866098873\ 967238736140344780826902891194219044838326368605979260746614906223283168\ 206718473812605687091690672414569298365327559625583438410497631212312675\ 116726099202747367331500615736749024340107524559496817871894436050591434\ 527702795281416164430882833582480567643342744279409078222432107008480907\ 429224625651272528587906365814068159539513935675528703178824621618409516\ 013761854850637871498127817906825375299276461127148550561968147718755093\ 143656919449987528884218160652350142484059184681125454169045476297956778\ 710341519622257774524839219406624188212532749474613923803822648430519694\ 892224520801909278430314674565713252044359642152230298505439023829358805\ 466349190227874482085091539501719356005725172567752328252611492959389996\ 105698915559647458966530933937548059654845401571467383610030086927463879\ 207258836859654735944087607561842324151201750319058959439953277691385203\ 338595492969770355049182304505660948019027565439447091829119654098335399\ 242687075325882891589031227454569605233871007073419812441927405853042663\ 254323972067402072579658705059086955756043382545032225054068184535827318\ 453716117330937990499235795467288298401971510673395584610113386934910791\ 442747021262988395841312293512959003621957792775644126675476744202550507\ 647595581906870832060525575694242886480460697932534773749282835441862585\ 508898534951413577149812514663278408994348542405021431571547029454573766\ 074725083840307423977349699682759735361645324020056675605527544150730914\ 990927665753940505514975606374457012207386763397942249754086281449600440\ 342905224950697728117895187175791776981340586494369882814199562339204440\ 401858215080069806073534378443676057640170468198414754709445121310201209\ 904876192939648260606615555724157640852251129391055461370063760347902747\ 160441198987349739479225754808585925745534144588693117644707890613477353\ 301697703622501583872941909960434928790389329309484022441523001072538413\ 259306823551638963580940096278365050419412407568590286309771495937248215\ 516098492598553556021512050952865098560718433219446377490852415887022737\ 441529237709018807413287689723281391196853413089573445564873001774233604\ 500331777303045706443853102785341220341804313477733515663446109507248009\ 705530868859096624675831772908152496444029275823044895420764275421648432\ 376886460934688027531741935806283047431104327367948212034949793159383778\ 148861838005294291474424877193930304114740505001060849883284843892342713\ 825399184820909753499362575290712382062063592757647039066686755421467162\ 296080126535475124605636856786349936995958459446946426772920802354028263\ 754852433300379002215991955447563305636974866731368544812385068032935018\ 911717144466339890944583095763856668842384319084064083930659050333517664\ 362997203853962792309329134111432919581908219759140600439728370096712980\ 655089557694673544127053726595967843485923846478102347413554390779038701\ 682185110267598974575320223175746598518939272921657371231634128443771915\ 368445667240288781250456962721720270935233990988019558607938317608629943\ 976341642872881876833625338711275370647481735171874287188335678418968065\ 651208632344339797993673928676065155257926999465706489844418242347354882\ 817465791178389826360820926216942013801107486520414991982913955914158204\ 791465905663683217970162138334790492347421562864535916969909274743339285\ 403993035097269429103118338048515174889879741205118030989384426767859555\ 839433111233430700960150000037584829715021890717769801408511071749126884\ 917345285112066434197633618090631408122209065201864790792096891105057810\ 009239203538842079643280768558527781752558598898244530781301742644222180\ 779255883820449008848457053559574079508552581031060787386060112532761933\ 319501990162405373849088848917054866544861387688191022746158138101124178\ 600649004945040005463524519450091190441428291450065149829760403460113506\ 187838676097961532487391193610014863427357016929005695375217833808549811\ 217359124266837596583879737240871373040843804183302149667967547929645293\ 023521710820871330697988910023472281550538443191448332507455957046769309\ 989256839514401882253997213804743388478418793760732983507439568715095051\ 176770905476247034191266222276110122654177228008805984035766400405268385\ 035764725882959398509249084963860719892686519237688425598323064643067548\ 974420018544547017532553841774407180449094644785527794839853209604479965\ 976718801718132837353814625637809965275987321339618349857700740242895371\ 630462839888707934165992315258308592553384598662727218611469368924645990\ -781 5417977963207027600600190 10 The smallest empirical delta from, 100, to , 200, is 0.6423825498 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 22 22 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (990 + 44 506 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 506 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 22 (-22 + 506 ) (-23 + 506 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 506 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 22 (-22 + 506 ) (-23 + 506 )/ B(n) d(n) But , B1(n) = ---------, hence n 22 1/2 n B1(n), is of the order , (22 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 506 ln(990 + 44 506 ) + ln(- --------------------------------) 1/2 1/2 22 (-22 + 506 ) (-23 + 506 ) ------------------------------------------------------------ - 1 1/2 ln(990 + 44 506 ) + 1 - ln(22) That in floating-point is, 0.636342887 It follows that an irrationality measure for c is 1/2 2 ln(45 + 2 506 ) --------------------- 1/2 ln(45 + 2 506 ) - 1 that equals, approximately 2.571479811 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 23, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 23 that happens to be equal to, 69 ln(2) + 23 ln(3) - 23 ln(23), alias, 0.978871131632306364 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(47 + 4 138 ) ---------------------, that equals, 2.5644644366606059468 1/2 ln(47 + 4 138 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 23 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 23 ), it is readily seen that C E(n) <= ----------------------- / 1/2 \n | 138 | |--------------------| | 1/2| \-12696 + 1081 138 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1081 (2 n - 1) E(n - 1) 529 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1058 + 1081 c and in Maple format E(n) = 1081*(2*n-1)/n*E(n-1)-529*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1058+1081*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1081 (2 n - 1) B(n - 1) 529 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1081 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1081*(2*n-1)/n*B(n-1)-529*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1081 and 1081 (2 n - 1) A(n - 1) 529 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1058 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1081*(2*n-1)/n*A(n-1)-529*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1058 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 5505195640744157558739419721796618644116655018970248520070389970274382929203\ 851576698309228208740358995823613919270149040266091258436850472188977870\ 274857695402516074345477747956041760806036078327160605438896749754412153\ 043223613921914230327875715570217503910712543619533956351849591944280382\ 226966438446713552373251869489880891799762784773477750330618261769218146\ 362463341781823997605092026614748967128115998764036717676563384748683257\ / 713353879780390253910592075962395558243 / 5624024923039691536685774082\ / 091828511840001936630321825120946746321232250488075148987406777128950633\ 316753455898840569727741414653317048998678177454389116723346459637113305\ 595779616388635750606211112318776926945050698656507809373301636812590589\ 348386452483174375377927245427340837792365926049797922332423869800749944\ 109920132862797432534052566687337139007796017232116949453631551647975839\ 990162709616033558245088906232173505112981263062908531116854691900891060\ 940389338392000 and its differene from c is 0.90117576081313345963657758277525182162778829124486186318438193122384822239\ 472846532226453954963101959273292914776460236158117101519234162280590818\ 928555253970191501212641171151721653025177381047595351311769688157281189\ 349646625259381747685127893402221986991142554787808671741258386812205605\ 503886472172498493844021039980007504288628824304016837201230254590340279\ 160149844762556689188007980814803625562024631727734916579914925800692961\ 187721032538038274265541688820172722248067097428087679486832061395590683\ 200358222241225245783574997893962397714456935044071625631140087456931148\ 166750645461396431277770751042738250794514798397207750978009860997778568\ 270369383607947841204200808619353579154877813998579446181162380014535140\ 096620636517210142173090479387281285515432543184676380863973683485215068\ 794963272713690679274423745812460081203260110713740916521797821271263224\ 065826099783955134679942079135629031118356345028235605215468965366777373\ 892684504803784198945947662716199753751178981116897790628044249806292444\ 075023618339392298284912815379128515748024388759304791161834822080331026\ 197131633730945991645935176261360211812266716003116678738142341692200526\ 510282474397859418885569928479368140428941617854947418861468887912588311\ 727488063986542220685951343870535551691271832578787419904569338262364934\ 045132400024062020745025653089125392236733361694471206621621900779555651\ 728295717120362770659761718737771495504452420217345427752472893335528026\ 798436279496614201150344658747139303508110001241940786502587690164503142\ 661172030225930655572214328834944517490785714055786855568869790623748051\ 725023225485415156716311427071755865155364108734247355972049578295100889\ 738077003367668372968457687628565885804778436463452519689699320855695782\ 878475581926714802714543403934034228511966242201597227772312418843379329\ 229072782591286201759667946707562593412155395172039433805580782947869114\ 513822235524651185586041737520914731638982197749387588354250020928955877\ 277575998828053886716899792588466588885544271467135479599637499308292983\ 406011127694782975625129714894512922812723964530288105731625354462281721\ 671585400668733336776683660005190047967782770006307300119749618534885853\ 759813462556747508139975091644729312643826721407073767717675952703009383\ 399922642994470581134014112541116445128940546233060644450823791432252349\ 350203583373967670160680565004887913563808547457889384490866731650101145\ 263977306329892195253191377614022625642759263653551504673706643974360494\ 889932509597898259504913260547370934462046873163293381752289617183638635\ 096189048211543048284796592158119517288769227318765610462572018947975011\ 949521117449076987591422416334989137935036038861721973501892933489826014\ 435506217913162059458846213809437836517373424276633459972000956374250516\ 208893379756368026993263784950861844626026424823193820435609015301195375\ 397478224513648191803736959503296416423628265285180170851821621607851566\ 023379040795346943741673544060471783809989199161206900931028292298947434\ 518607911626621377553258156681725056391738509202771210497531782984298355\ 594857659929423495771741272519278110999531731968599383404638584660661561\ 227123366587451015681024830263115001861508059805874178334046226513551951\ 222598906802403778985787542661189243949106526873004177545996241152063994\ 821740214037905266448088981891736996924868079926210142489706041684138367\ 749305942133390057020167207771115026097058216711538515881569770166839699\ 977193844618945171520851046675634494034361873805116319913007085976316902\ 330926177603466197511283857222476382144496325630943779628929611538879813\ 928094532626667334919721486734127848296296306980148961340806243683862303\ 635837057270156683596804595994487315235205124437063537148397594295739751\ 481986213056975841209164006056230800779778679839938259524030003847687692\ 707839088663261144378324283427041721323833300976665831597186237222005625\ 808498456907740025198965037099239932021230183684013829686343196269958876\ 248834452947551853384856710440271726546458356344432861658132560632473305\ 312801093346093493821883643729751045057408930529264176067739173854598477\ 363908928474372709814336965174479371111752434160897206249570672462649840\ 047346016161989272677313853696060278918993490442160240828175202928455950\ 561666126963145941008821710202316232997920263778300326197719666432785046\ 631242566487786847175431847333469910477202171665987764278902599278960221\ 855433667008336186950394936877606595575098553427377087972100701418065850\ 515993077743587959262707580884729243975911447059043018706348228475539155\ 482881534437655883802744445920724759181392089978728722691278857065030775\ 597756876426666132716509758506195773472161843297501982930448693416565624\ 638848163940678749889498213602417951978934432853712080021079110237277396\ 839787199628014647005433138595316174847488679953169253466220824709790169\ 477880191717065746671892943465384034284258090939005113294598835386966595\ 320257421808129512352413018783352420863446656886404664242197118650273909\ 464227692219290734396731389750658316045198486691129507423231196782075960\ 380212272527711594943581089342282517047541131066071065074797439679827204\ 800657184396292128392490388008018110376234686443303772088998816296730057\ 018986239237424665322340726851252523574686131283054634070746228924822541\ 172221138542287529708608379340906478841870395991628158502339346833961495\ 042704779295384096762127146669454002144359409019104250054023564171315187\ 953777163891632816880884623890500184072859214646687596936071791923035273\ 199528579419941240362460957258873285066197892132576244237811038531306665\ 655337117278880533013548311505092444948333865028226756657268460610075835\ 262695421440185669728566819683285588657033783463232269278135642869307532\ 730261480150604265380265745110535459544566030700231681738413631441263256\ 251155996802870854807481025040028256278721588934457129713253601746215387\ 916294677179588687188918458182625423190344734247966392715518733400307809\ 573861444569786457735457663366239372990215374261952858122795097635539850\ 057692867004534668696980819157192267876726769105445290692844326053888412\ 720218935213540848911673823645570446351525118146199573636897309546692384\ 679151989954590681987867036542761460003538454676857336583989706455281498\ 306437363215638715772382223548422536983391469928222818943439000562670050\ -789 57593818732803699 10 The smallest empirical delta from, 100, to , 200, is 0.6527855968 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 23 23 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1081 + 92 138 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |138 (3 + 2 2 ) | |--------------------| | 1/2| \-12696 + 1081 138 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |138 (3 + 2 2 ) | |--------------------| | 1/2| \-12696 + 1081 138 / B(n) d(n) But , B1(n) = ---------, hence n 23 1/2 n B1(n), is of the order , (23 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 138 ln(1081 + 92 138 ) + ln(--------------------) 1/2 -12696 + 1081 138 where delta equals, ----------------------------------------------- - 1 1/2 ln(1081 + 92 138 ) + 1 - ln(23) That in floating-point is, 0.639196663 It follows that an irrationality measure for c is 1/2 2 ln(47 + 4 138 ) --------------------- 1/2 ln(47 + 4 138 ) - 1 that equals, approximately 2.564463737 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 24, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 24 that happens to be equal to, 48 ln(5) - 72 ln(2) - 24 ln(3), alias, 0.979727868486123109 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(49 + 20 6 ) --------------------, that equals, 2.5579013230918975978 1/2 ln(49 + 20 6 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 24 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 24 ), it is readily seen that C E(n) <= -------------------- / 1/2 \n | 6 | |-----------------| | 1/2| \-2880 + 1176 6 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1176 (2 n - 1) E(n - 1) 576 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1152 + 1176 c and in Maple format E(n) = 1176*(2*n-1)/n*E(n-1)-576*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1152+1176*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1176 (2 n - 1) B(n - 1) 576 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1176 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1176*(2*n-1)/n*B(n-1)-576*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1176 and 1176 (2 n - 1) A(n - 1) 576 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1152 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1176*(2*n-1)/n*A(n-1)-576*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1152 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1617153294525159527372259180782097349474013154253674078160229658315362105560\ 751899765165459128854695385229114948059920900732615379234459540528213013\ 917929625517529525753264451509975086552258845519492030275434322453660717\ 304686067158963274596964111606458531567755425138584536584705385820571092\ 227911863838567854837646557943345795340493677184468492029358722267414633\ 349081554326726954094977753577355027979397782531215831829706635171055835\ / 78155026448473550442872022053748954398949573 / 16506147743086935861382\ / 902514907188877044852157764837596272871538431551154509330745744753774153\ 746436912592479581816375779704646598707816645014811591236592951725568700\ 317061531472860851800335064393877752740726953107923468559466872343462509\ 225653800568423617092027285371372125637781389835271894007758582622646226\ 388988744552733546126355027616508674699066152993612463134518968679825875\ 087939946434538689809328402401660177699457635166991196636341694113078793\ 2368773966896194521263000 and its differene from c is 0.51799939260831116731241709550142197698762136722038892341074989188880117833\ 015590574315358085605298058758033168348348256374260406872860502317133286\ 622189778835048844410888400742949023498342962832134100750151672171071613\ 346355194290339844893591982936958503440865634228672899063941785889720515\ 374572026180573827445552781074052542736592026450882254266496826974571552\ 581056232557121766541888334089129313639227758671810813915853694356666687\ 364966578254552916104099102870284533162598229418856687883643197643417734\ 979278118077290988993450732402419736254133227698983973841364118264306121\ 617395594105848887899002403484725849607065971485964876900902051184190042\ 543031849061961945259464118784587402127624947766664505972688616949867996\ 522084944792540180817828852548465835820826800344844618074966185436323726\ 761212718037291191536061798109099301759661183549663304161452261562809423\ 272668410703096829540958973669690344357575762701065643709771228840751256\ 120802382141051348760262509903328439972517472539497167408253642404772901\ 020885586592378813636335503372810725223168584217679456133925578069794002\ 265491834667513933333511751954298746137293222944403406780658994270130254\ 205023827807675122771146911050159366920948918842955528125120463368051583\ 249078995776504366448549230776318110364972620121198845589442229313252824\ 452889728221676889180897850486712802241190135640772192220112093872156095\ 140050368561138214782689431627425139965440904827050297689721676538221798\ 331779174422499524060549020163429607492270782018345784981328428398016200\ 448090291898495801743864915231446506255580557335338961168114409346754851\ 317794310291318728467638422898566863784114252754073487859417317274601705\ 830830427312045383173588665095859541802634745872441793250202796984768129\ 026649723110421581881304095047320109606747703422512616786283595506360633\ 415337274315677827825802003202758579398745100958596891853119048043415961\ 232713798011394826218521497612382755007061921293308744511695506385086105\ 251041642277553487700322143639954981053221459926616243759309873266986439\ 927864244428029336036468587911864697243030396151123697238033416717274538\ 744774106836951971324479288782558670016179764797422154167427331872746939\ 414452694094380111626061623966066790430591684074510727344983552201061408\ 707979158676791789653500706341080186739095840706090860350232577426400634\ 281565943309847329940503711277837238056504678019039721984722057977989289\ 004649293213898452610325443961953176220904518093138831877082559373245825\ 237038537862601745261981834584681841642960106413079733654740760654772999\ 834095696057279737990147445725204118694995095858660709965742651690446349\ 519870404796103930681362901783660919563753295298652742662672447043263162\ 600149547441638005853191365305866007869740148213836092425825974905503083\ 766469391571427778297517234656710220153993967683961964599544347439292233\ 445883138011797434630913537174876841412095607383102871628201072071721968\ 422231333804606078630122640084279337627318099197855393412826880609883661\ 986794591045000689255496554727983033461189384510344009782813211575216477\ 334115358600775570751181903176096140053226809874239527555899379016888014\ 499325688587893346801129030565615755033450718877379027727836412285878217\ 935967833660227029587930261629593130940466278518711136552774180162177146\ 151488346913777261242233751023196628170696351053748725974811531125810452\ 677712163385892237473606561529437246679977603492985384695672301042767862\ 674678564216361842733902965133700774311027930330285543138516595646722808\ 879271885055757969445881094714285314234798594938871788793818979627302985\ 048243150889933857549158271562074920438372826062003786164835785165208484\ 741006772680978767095717948452869723038583640226149199494983268868294432\ 108893238678711613853297292411545712240875125793325286211347210523925820\ 604252161456901109175777677867789849444499425232854547899893030351639268\ 762763216354304728010808463927565311723683339504055250448071043856794404\ 004761936458614546530065517003726697022507782481993390229952159330779153\ 043942651063342992385223844221899531841538657074679063321185645584671259\ 542915598674829173319083753718423243101777957651343948632361053698516192\ 329587281189254083152099838921904056567859715957370065133575974091380420\ 623886544270859832757633862791035171834907156882523283303780048736883933\ 096593507283236425997255149694303117527128602307151562157212096860381341\ 682888374285907666915719764733595861559078729071033900301158456617077587\ 705383447670814090183041210890113384070356577682903883648257305517705243\ 992046746163112825228537420742793177737444788474588665656270485276160099\ 479235194763735624126411259528509847300981825515540871597814454473270912\ 470440223090332723253261304928933274853163250780980739015778705292061692\ 628641917794570968444108457557058816301906988362546265322021578388579373\ 552025704261255043059531539501719089732401547413900395462030453635758500\ 055070148811615768620230506437147014385002281853300628391512275040720987\ 667067364164807415736883060490915657754809900640489513348718324910462054\ 629077769920409661040330441723375609037123393257437430381831240499178215\ 478779240923221541570647667513263626014545427835262277159817550705821274\ 693048732958982816283452923313790196388402452323189850663524397698779690\ 664631429659767005713848212288691505182945042630363398467746047304397945\ 510687800294009171968566995254801028415843968005183378043578666708563914\ 172790321629302103082417902972476082358083999760491078742986701153843279\ 073065556173556734765079503664939105194380159173651969147186329226063359\ 990599638522093354616619457726731762515662518790668158712599462789666699\ 119811542326094881569443832551694588200675908211155415368229408871686559\ 166951562006470964461950942468606006036519096304121292104933593500029710\ 718105446675450098555296626041512964671843750532333350332828061031058060\ 016279580815088562760149724661115906562825210640611492226631688659563734\ 057137023233235224059354176410106141752161779409995164187500960648918997\ 074489357466516730040230739096215947102185527136843350715891161379907226\ 721983035546579745735225154517762604832425431962434321541740711384136358\ 052280240485967871520798172468869194443601300758045275169775506373781404\ 346951755779565417081412913288944822092909390088009581227792430571954883\ -796 1229085786 10 The smallest empirical delta from, 100, to , 200, is 0.6481287487 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 24 24 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1176 + 480 6 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= -------------------- | B(n) | / 1/2 1/2 \n |6 (3 + 2 2 )| |-----------------| | 1/2| \-2880 + 1176 6 / Hence | A1(n) | C | c - ----- | <= -------------------- | B1(n) | / 1/2 1/2 \n |6 (3 + 2 2 )| |-----------------| | 1/2| \-2880 + 1176 6 / B(n) d(n) But , B1(n) = ---------, hence n 24 1/2 n B1(n), is of the order , (24 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 6 ln(1176 + 480 6 ) + ln(-----------------) 1/2 -2880 + 1176 6 where delta equals, ------------------------------------------- - 1 1/2 ln(1176 + 480 6 ) + 1 - ln(24) That in floating-point is, 0.641889033 It follows that an irrationality measure for c is 1/2 2 ln(49 + 20 6 ) -------------------- 1/2 ln(49 + 20 6 ) - 1 that equals, approximately 2.557901676 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 25, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 25 that happens to be equal to, 25 ln(2) + 25 ln(13) - 50 ln(5), alias, 0.980517828832032408 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(51 + 10 26 ) ---------------------, that equals, 2.5517429074567953112 1/2 ln(51 + 10 26 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 25 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 25 ), it is readily seen that C E(n) <= --------------------- / 1/2 \n | 26 | |------------------| | 1/2| \-6500 + 1275 26 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1275 (2 n - 1) E(n - 1) 625 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1250 + 1275 c and in Maple format E(n) = 1275*(2*n-1)/n*E(n-1)-625*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1250+1275*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1275 (2 n - 1) B(n - 1) 625 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1275 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1275*(2*n-1)/n*B(n-1)-625*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1275 and 1275 (2 n - 1) A(n - 1) 625 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1250 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1275*(2*n-1)/n*A(n-1)-625*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1250 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1058220993797227377613036384060358008042862584369927761891829422249600018001\ 945547627170200231746782176051542204081256681814836030092692587367386116\ 266437857698332635860674294664316984267126675940188636223183022639917308\ 491061921080737974020740241986463072518212509030700885175397195085160068\ 282879949773389682557860969781458216239864229166302069739167639609429438\ 437718294029191271964493544279598519634160979429848956994697968549856500\ / 921990409734864587241886884998739268246351593 / 1079247069946451642232\ / 827796585421988703800198940472747809759081223221123531754447169825286365\ 002569575692343157998068430405325528849000915647348961861796621129979592\ 053973400303917116887442189615349111377030127383869825534798105015052739\ 669199180559960602319032165056042799962668070899974419698462179473920180\ 969949646066893717183074362916175728252956422121552630248988335073985742\ 113804415488936230667808942873882028460059932656374693595809742483957485\ 770469310551574502226672320 and its differene from c is 0.58029756928244650616146702983242649824162041167939609655592092938865232534\ 720584525704631789046986063494295100382389182226216060542385333105766828\ 112415154372971333011596016026576137020807294900038500057990050191956757\ 656422683061338770391977626357287950256959316880743815395265150988353733\ 145453908781687225038261765701684958387799488836369340326231119220734686\ 058786783118636965916825886410547114395147624114246781704028940836261609\ 741239867733120152232143086439170524500895775858114679992282485834935825\ 889270896589541069149654924714432640484376377840152906050373154503360703\ 197696415467693835308583288183478781763393291361276945083594253706741091\ 090676263134825373833306052821267085459109082563926483299239130936787187\ 269732884243302740484726935453946779685146050299913285237338658351871896\ 773667105740940885985026286125728325460079453811964245616119332775211925\ 252257957700415437113686409362509737946632492807482157264446033679681643\ 423805666743964217625000679991124066168809988832021826213922424660940546\ 243908347555834539278162070880031634393075639037412668451532258203002612\ 927524951262873284019470726562675204366536577692246574037544822073978171\ 041073053298529198444680391170602090232332047445185857413832674214661379\ 508242905174254543366597719296159076210443983639761802448241657035531102\ 246553263275670586738130455658163580208733554508176891105912254510212388\ 061999265704083504262345383504870004715175595570603662135940892348511806\ 128015358230713043452236307529030442271824644817184681346888901032760485\ 325862914548038008499242572312142331512607403792505640915655359240527219\ 250043284348265384599408217900978508140114667561003249999340986081746105\ 311362630436427579376630796293275345294471113494984855816771360879934243\ 578272497096352710341165364699912868797282383542743665275150768163144931\ 249822710932190215610942283226062833736092679243469986812074616738572412\ 601978386545845018639088265280525965738131651152969771600068693898941361\ 485960619597526817913533665234899974275880388559323384512797488484367851\ 647980854603985203927366442552485467210842747440065500992306030264918299\ 254311110951142545842621880287536773332495037566679011321034304990750329\ 665538641744106254140497490319613020619227172751968407470666352323019186\ 287013799214062138807620082224050334885203264604939694645517486242338466\ 396771292916435495143958513799727176242240886821519465887525266095848853\ 269129892678422003837949076333290432254709329119204948079960266862894259\ 560718574520045717128478841083127138472320324762659960365255099116359975\ 449566609267601765174120349155457423664638557074644227700296088458695359\ 302086766749920511465882248714358700040134344472350070882716308414288990\ 807658358551260534371042233407705984450718878929659633766038088756272875\ 461574676478673220310220956705038577019300695804773950472539684976796323\ 505160797730104572077225242115316488648153964807361603963597278806349694\ 640405352380483690194727175102629149825576481924272597700745237537342074\ 969971220720314891666640512020051007384596766271433050391656658769660323\ 141454753215892614829815953880457308252999297983598386156550439389665110\ 924277165880023734523787614385858300495444011053661849647525767847845371\ 236862988608936133767297786983848552699103942880049761015340669406630587\ 936631423257451007441628909569527584990496436336581809007735098198613240\ 751223910839308590825954456873873111283410903239928628366143003208535465\ 414768605265137385074787441299527535120899531635165118445908050563776830\ 893706740209854495363530947667136756254737563688895946707806078601858869\ 093644929371579934966330129494576117236914337228929871491245871475268505\ 830468856662049911476304333329789527256004820672059626057999332258380698\ 995846230285444642673430753820010353492038470596611543605731823759120236\ 277675554874967108944380082762341188892129001869777958892024548221036904\ 833282519428458399732581775280867548202173191805975619321977491906797397\ 710849865920419314326321265660341567013205662706123156775372619311131338\ 814058583153048942800229131297067159358951214335077733362506442698899620\ 445534693443745922607939034936881519079739565175912550983434774568001439\ 199206388691391525790164464261199772111369929230994126080400675883720152\ 724379117932126629731143910102952286749404890170207650437443120567281648\ 684910310670573952856575597702318539494605434258462697400239298527322085\ 544221464757491899956426142341262300714082803960432392759267338396453815\ 724250975589164248895571937507074221055879773727472121012840865364959065\ 927882408330402331497592965854636984551954966737511797881082008151863510\ 463062210323414040192149134692488910963604728143488467180069393353666620\ 512021052469656242090697344390277701318732157921275942194845249577016585\ 377065727435303453559448052071689242040218382810005503011735581625135357\ 160876767844670758643604543115267519278982641541705171854114477278490523\ 793931119766080846553941982440960090514367985422437967599085906349963841\ 975976678215114668567991939334373833413627955736699693885213749857812956\ 785612920009465032200570196086522955965697817100900690900920361983552725\ 203533791386792109581285660876457606432638293355608466540677475694044942\ 337789118801207272345071846531423932859987511125416023932555409808196317\ 853737526389376057533056752338679451634320447991422266209096316688770189\ 157851719649901376583567873018235777596926171742780754932512858639280580\ 391567530845392716037242469157972842892077764076387647038465714628662368\ 382508273932445447084186552967129597411253692960032320471115305233315460\ 205014633502233363910335042624295645708405051280481582135841121803065431\ 842180874344967769845776175317518581857089130353005433002500927613235543\ 450458217377829333033348054980771339787710514837080960874249636714505237\ 231098462628810837148003699381708525519937896784786074698527261625151375\ 345498624775014455294746014763942116751005594099595995183057685086092182\ 328035959998895064795988157033979421385780776534128360644079349236335354\ 070409123235189549867601677478828477373425255352206761137264951673014341\ 243670940192349065600153592601908146662791694437786509969713999622188456\ 593230058111337542670639482075093409761807958487912985921068917590849785\ 317771694894660470297587812317308800841470934873576053115274707979630729\ -803 336 10 The smallest empirical delta from, 100, to , 200, is 0.6626113512 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 25 25 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1275 + 250 26 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |26 (3 + 2 2 )| |------------------| | 1/2| \-6500 + 1275 26 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |26 (3 + 2 2 )| |------------------| | 1/2| \-6500 + 1275 26 / B(n) d(n) But , B1(n) = ---------, hence n 25 1/2 n B1(n), is of the order , (25 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 26 ln(1275 + 250 26 ) + ln(------------------) 1/2 -6500 + 1275 26 where delta equals, --------------------------------------------- - 1 1/2 ln(1275 + 250 26 ) + 1 - 2 ln(5) That in floating-point is, 0.644436625 It follows that an irrationality measure for c is 1/2 2 ln(51 + 10 26 ) --------------------- 1/2 ln(51 + 10 26 ) - 1 that equals, approximately 2.551742966 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 26, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 26 that happens to be equal to, 78 ln(3) - 26 ln(2) - 26 ln(13), alias, 0.981248527554022745 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(53 + 6 78 ) --------------------, that equals, 2.5459483682730917369 1/2 ln(53 + 6 78 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 26 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 26 ), it is readily seen that C E(n) <= ---------------------- / 1/2 \n | 78 | |-------------------| | 1/2| \-12168 + 1378 78 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1378 (2 n - 1) E(n - 1) 676 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1352 + 1378 c and in Maple format E(n) = 1378*(2*n-1)/n*E(n-1)-676*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1352+1378*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1378 (2 n - 1) B(n - 1) 676 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1378 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1378*(2*n-1)/n*B(n-1)-676*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1378 and 1378 (2 n - 1) A(n - 1) 676 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1352 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1378*(2*n-1)/n*A(n-1)-676*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1352 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1072511953364267441414933196347005056336644463670080836512172622020403533165\ 490114812658266377246601858319469651766887796694050019459173912412868167\ 402370136470429245984638888670827614750798986984457277173513590962310620\ 227829331765206943508397143942677499262564332103909163081443474561877776\ 373100300964924158930388064295934212741610923313768209357738673582431956\ 859333011679805948131334516290920632996595983952949019138239074063722652\ / 072492514417759896385957878356030226478805686040999 / 1093007452492936\ / 532623196147991125107991262728485280302907174514389734317052687173182341\ 688493395839015993542618886046270225981525234682103872621157202165937775\ 350582358355618718378223591474666876700842363615601210467540405595801822\ 717198656898882840029730246429429125424413410069002189009657081017587413\ 305223187115590748108022710678156197673910240903832372404790714434501744\ 292999991772322182124678202720076662045467562078113517510743816103738160\ 109441941009408840168421164865746412000 and its differene from c is 0.12035418001447235226658824690198080308920942194490298771751586862900328068\ 835369472496222419109704929551830435444130820391821339977508483019694972\ 572936620512405166068088868958167240093100137702925227228002733502983319\ 366288314871624586645122831348189156019551994983303421151290592674740349\ 520707514366847898681909560167388919805907156274231129929544251045283711\ 926961541784224720492026662920076798017955220750840061051676954453018652\ 627662344975481651092157991511516015953421805627172201046418779731789171\ 802692182312293439423884437938996648348365493205149546900921156683805682\ 668089722618125450116006585928843796818489507960124911223399509814011585\ 099939940152950946784782201189434041055611265467154903962253413221352639\ 284001260661747672849394985174316860489190504982090027998241704622908529\ 113830497038881496222295642191662805725774623736334590438808841734410954\ 737264004187386724213580565527336832389578200165445219074631759369460509\ 175326172095749620982976509848538982060370107680944721279113590577837839\ 968931688591109219458094961537889012296918483142859026352658756565294619\ 296760935178136670539290722790504640189229713049484480349428644008519919\ 760363831269224110441812723572018663354047074854111811787591665794709881\ 396311296350623200954081077416418693320256996518439402451908564259323684\ 589620825935659582918576918994515215797926344206838863613893146500547071\ 365947268403805234779492685980919902374437577148675989324588245532955778\ 835069116112311878748706417474316945442534673400216096063970762893368663\ 217069695630391627625567679322415191126161457070543551867747065500804697\ 026791344888252403984397464141006658461483342591664088875681136081188184\ 363335190233224339781286170316331113689384095724966858491762648502115819\ 533527949950500735122824067445997110136854563606928317668037270233574563\ 630970328228099766066586322447162604980787232125805001091239089791250913\ 898871697883092316915407063379565814735968478290306497722836609234751586\ 249416655950003286606685041275222093946967717658478786605285892019811950\ 003005736375923405527163608356863178138338386242859691416547418308622984\ 697533692473509746030801560762577790511234813418368025588016474695301385\ 285333320978721973498754251519591848343386778833469516146672207278183439\ 221490345702073621734959340282107437139797202425601737792980060891086492\ 978453506996271024756911109737233256029086475285895595894752597899759779\ 305401085683161600609307686425518950566570567762878530019937719312198676\ 958602675492549048469751200897855797292163959936045409539163906521594652\ 332311711400686109989212245331674098138000602208214776826362813714965080\ 307579417412390489542773231375510114709439667953214707919702441074622105\ 409944911560836891464096756288653863160216628489844367114099652836181929\ 251705755425230743745567410638555825725181273698622089876135734936752479\ 675370318926276522615494413674812662102644537274336266142542162460735849\ 154949338132575364791236648503139165611585392089736683034079128620726759\ 045454148104747678274835821855956905108916851004312605485515042649976345\ 548500066804546210920143453882237052864978662117398272241073776637022748\ 537322372987183873643413775001884451103063768944544027845179093876737683\ 752035089331373181412490921592063006261938727099762796552619900461006280\ 656388998976098106880086164401711140809531244152313673627387359745443928\ 807777876338031517912702767646767406369926166610130535137870595401119872\ 925815359127360122690795088791027640504470539510254224821162791730593740\ 455384949405060262164966050345105326765876394906498349632504818068228326\ 709504265077016528768550830982128158744330663628495822030548975554691473\ 454619803735085107982039705377166787228080109933208147289845226845258808\ 554384983378909796587556297886315319984373352715897620141099873942472542\ 848000144650018366369983107734165308170853346001912569227055284879661747\ 347239259670688457210474883155449579555567170824038919707282683166626245\ 465011964079578454723939644883751203329572893679091227218309437942546912\ 583682010139399237631082241053198104813326311919130141348324049063024682\ 175243911435494537625902881650910665855098440458589749067135081869909989\ 282961486839286452251837705996482905831001973817319540958985263275447624\ 405299783434267346138139785726093787501956751145430522367852537155795048\ 951680935238575673975928743647828315356271409724099176921233240582368938\ 809448055775893014533255613156897697916197007684588754671457249862313386\ 954015826073086754611970807566104377686529674489825486448147941322850174\ 980912211432641097068054852192334917078478351377821707143339073212911659\ 038648058145399273401747404223078442878744060291228252994363527165517932\ 585532874565619603092226081394963684356366939464475348855071096101195916\ 347215101086951566248295543253305588241485256597072735235916076597105893\ 056034307515128067107729701666190360735751876404089014865179550419090904\ 490375471056733713873259939858744747952718212375507930727191808926547927\ 960921578711427418133528694814320307248417823966812748192635139670162980\ 680952801517086602330425271973220310480486619959777041643493765500236619\ 962380120146316855752479527378899764715909128390545710593311202238318722\ 880624292252900343087646845112030980012060989498175080051434635448581947\ 907794753916920564129754146402449112362019941571928154243300937133764032\ 049112840584905621103349091166687452651150168975112912971505880340658844\ 160039377569595372015854619879715701559777865112342890587335880058935403\ 838377636453776996961782353824532833073573903015567508667286116582492435\ 405867826664400677181321005034611650780388206269383548899013626046678430\ 047755547018537807060984814814469244320768343517891061350214415715860616\ 060247181979473568885117535643872748939964370700027816249234468934973968\ 989724759341317563104011810067677669352538632156130833399688333004402402\ 501966701734474507274521695930170054727865090218810476255301492165950507\ 970849625063471923561744682089737510155032506228529219644689996266342587\ 961301287281204201669186321547577630459170009459084129318065102522353383\ 915168669557023759605132862131765726682297864567976097466450625184082672\ 462654589752490921578484475186153671687762245550816486847375002151795875\ 783505029208599567419672996495909949431853107615958503391906212765670 -809 10 The smallest empirical delta from, 100, to , 200, is 0.6567394041 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 26 26 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1378 + 156 78 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |78 (3 + 2 2 ) | |-------------------| | 1/2| \-12168 + 1378 78 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |78 (3 + 2 2 ) | |-------------------| | 1/2| \-12168 + 1378 78 / B(n) d(n) But , B1(n) = ---------, hence n 26 1/2 n B1(n), is of the order , (26 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 78 ln(1378 + 156 78 ) + ln(-------------------) 1/2 -12168 + 1378 78 where delta equals, ---------------------------------------------- - 1 1/2 ln(1378 + 156 78 ) + 1 - ln(26) That in floating-point is, 0.646852443 It follows that an irrationality measure for c is 1/2 2 ln(53 + 6 78 ) -------------------- 1/2 ln(53 + 6 78 ) - 1 that equals, approximately 2.545947628 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 27, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 27 that happens to be equal to, 54 ln(2) + 27 ln(7) - 81 ln(3), alias, 0.981926392613620944 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(55 + 12 21 ) ---------------------, that equals, 2.5404824443868363439 1/2 ln(55 + 12 21 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 27 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 27 ), it is readily seen that C E(n) <= --------------------- / 1/2 \n | 21 | |------------------| | 1/2| \-6804 + 1485 21 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1485 (2 n - 1) E(n - 1) 729 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1458 + 1485 c and in Maple format E(n) = 1485*(2*n-1)/n*E(n-1)-729*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1458+1485*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1485 (2 n - 1) B(n - 1) 729 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1485 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1485*(2*n-1)/n*B(n-1)-729*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1485 and 1485 (2 n - 1) A(n - 1) 729 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1458 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1485*(2*n-1)/n*A(n-1)-729*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1458 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 4233834828729561171098366771426029367669873638524426589113076341505898373566\ 234989439382687860444406858317001594688375564254085411649007811167848538\ 305108140448519853545258656481735865231325322653335526176613229566078022\ 967158390046227991775318965303157751629186752037126715789434150651829668\ 640344864513769773016641821370663263619183378544685815678894301159506563\ 149692863478561580142368585050903915059244635073806601748586044549343447\ / 31560771886819395284387259218169882317120711422811 / 43117639576427359\ / 168722146869864334271841760974593753110725825259227557834350935686505234\ 755115566661423820998932386488874630327819147163478946112903229921178873\ 586420340658146133105684705481704453868950846196036231038592550287024298\ 031886915341531449270008248138809952704988927772700025411175191273525758\ 292572664604486518935297428700756706383152974487539683552354899110438673\ 548981595880754951673092627420080293066184328054901047589602717930723285\ 0999934736452320895041574312456498880 and its differene from c is 0.44149907683740791546165489443167110565825712758767122629547882049702067621\ 161944750182317688479011205564712831007543068882663048186814872094323184\ 767925113248360681135779515642556105670066727291406565388349882288772686\ 730579428507115932405948340792908581757730687654193308103165627209814388\ 308129485736723396822367020962066039958141559792671378338444927954577121\ 176919362943843096225342160248993356273658929814729838511457249444803317\ 825345451925495350717923316154625479119070239721668696168805918069321605\ 860628528575184411937502037984983991856027808602571586502223625775756950\ 255681767673905118497205477856305426685524362115259748027851269487134574\ 364831049686947351729901171291292474767845649460929637825383508950366720\ 657469274828940845639209595653102209240826830100227802741866375790512132\ 603547509392124109016934098155844427845168173140425445435307771040019866\ 003969794148139937971498429542613000955485750263754483672376635159974529\ 685111080414523376186467787006997158837746940285668142136762559971335866\ 172414628501380611604120037939606282326407928522673624181325736802671319\ 224102227719731311490046968734571292908725120575218939634196434292747662\ 457431086534368461799411397896404926866090677554338972353985210610039564\ 917917133662394160302475945711178062669357327797330950805901067125469755\ 061873111509464201397967855803604859609894165381009282585511994211750181\ 423282012902246766539719076335947741371828580238319631885880290128506957\ 278310894994777874931607083008041770471450192180523318923828786768836053\ 951673097141647362904739624048515155286147435986942638391848686636835680\ 247900972713331337092502951714667550801952273765829147253836390653778134\ 692445628359728472491760948231872495469194599555803293666141437083414941\ 462238543311337220122127295918965171920922037435100980424256273302617303\ 054053229170056884959406835521709692972245148463601346727697597008258443\ 172910353941936963537494621404784333494371903643086088791697285172178679\ 953690321694723279030505090571157057868665232985066887959495337609120368\ 644478707463342764697319770527846182350842376330758961503012462704047217\ 692518368984696118196901429507191617797947823504694764005324177550152913\ 083322571725702811468812709527488899956468782129767523914093925587861348\ 867361360278558808991326798166853206931817066922463187276063423903267729\ 818429930602086164796808292439708147932938280392255117687852212730743450\ 251315470244002818733036404628840178800666245350360321004610880662183534\ 254309531187109870652444107370865349923887283767711635653576308097663044\ 521686073332625615764515496916407283437263007184476454037540185549020504\ 923637189467916869183910490142806694039895489668020238380285262147113615\ 240451879752015586181621365953482269457770797751728051646439334017016671\ 384334880839443191295090256763657817323311494366829817110155821583075611\ 660510797982769374937583985433361442039194472501019238951650741394998392\ 415695853912262672727440172162456533503102689058016544707752169164335608\ 027744164678819688507573078041234591431707996670357595575591135259866047\ 685652552398397358469073787041180457023875702175121686779762025671955300\ 160919063245932488307557936796291372713942864126967045825611231106552567\ 333206440019597141351066549047469554936569115696150340785767305726442506\ 638788728773628681552436474929815890792740162709811567300950694749983909\ 950752735432320244500192888416712429833083480740839409942936666379488552\ 081948638805440196219629956623197700309021082972113273912718790314755896\ 591164192053589042767043546733154137020293314639783369431034529123179087\ 593602479965330739208284822247503168524134425476736276738970304339276222\ 897437490263622068976006127120585714439327900686607948251096258676028498\ 276451870033366135717642084471188355966105073718543870605532727914927270\ 454002740168987187168337443796827888605779551032342880657556310827817891\ 875878499398607855248891120386656479225122544022157043186435178078912206\ 950611184486137493149660290696786738500795124189113489119801990659806150\ 233382928140439946930333592207404240216962518585751130454545927747224143\ 401366427288828107655534626897279001371496428848381528681381534083932181\ 254651096961284315477696722998484139700414527997755038252673346583881259\ 652548496852417368765452441520884680194498689969454121475360124263599799\ 321136637263948145300227548393612126497254644216257613760731199793548681\ 117484759872818465828333363875917088003484474554517357583960753626704988\ 513919556179706769685233086592847672767624238532527233817712180788109183\ 076510302395854017610888751516379368861328625036594923522566980218632940\ 932461063550620134830591736151017428620510487471731366164525596327920690\ 596170660386358206222586696499283986908927668306447436029726432881468423\ 156938976522657750139869940258334151187618971969008226809296297500476878\ 687942383360706611918054834679028239879688125193453757729849922277636840\ 048139965328006745828262116054001505461929883305417922492136460471129504\ 325090772997823919366569916619828026235765098898724652676334466430892636\ 921340582424835565156693153642022798129511997495923681615038904932902467\ 235709628825746772785183403930230049981087179926064211812982727382991610\ 181291265578116039418042144177835166348153092555333897425937136557294213\ 085342151243832426697603704926406689728355050754442143658201494444395089\ 504295225825269187113294864991579215680312568888354163258375327982137988\ 202635052773005758295494016406805204749272376555721375047084398699625400\ 910286086289049866749410353667547848288817525822975234482189698006034617\ 248419359304562740313968022802471248498734460459323163311447436194805292\ 255743193133833401178896394223070467787155292616563073900243268605417472\ 831725469844295540327316244363889307501049218690779767693439425206933788\ 748036611367006037625982761589520048838658182717977784611651186992637293\ 022347807477967021156429266394816835079941383491121441447974315965046373\ 838968944170903040874213972320714930230970245567920431721809262848952780\ 801675395542644804630074394618258770609306643400515110188889305658986222\ 295266762429860010810337115407604197541939518742548644477900133808921427\ 050807648160420323569197549063490531680071040289014630138980921424030316\ -816 13292425065326887953445044553404214792140793279321363626957082 10 The smallest empirical delta from, 100, to , 200, is 0.6652104786 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 27 27 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1485 + 324 21 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------- | B(n) | / 1/2 1/2 \n |21 (3 + 2 2 )| |------------------| | 1/2| \-6804 + 1485 21 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |21 (3 + 2 2 )| |------------------| | 1/2| \-6804 + 1485 21 / B(n) d(n) But , B1(n) = ---------, hence n 27 1/2 n B1(n), is of the order , (27 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 21 ln(1485 + 324 21 ) + ln(------------------) 1/2 -6804 + 1485 21 where delta equals, --------------------------------------------- - 1 1/2 ln(1485 + 324 21 ) + 1 - 3 ln(3) That in floating-point is, 0.649147289 It follows that an irrationality measure for c is 1/2 2 ln(55 + 12 21 ) --------------------- 1/2 ln(55 + 12 21 ) - 1 that equals, approximately 2.540482440 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 28, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 28 that happens to be equal to, 28 ln(29) - 56 ln(2) - 28 ln(7), alias, 0.982556954715562891 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(57 + 4 203 ) ---------------------, that equals, 2.5353144976351150421 1/2 ln(57 + 4 203 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 28 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 28 ), it is readily seen that C E(n) <= ----------------------- / 1/2 \n | 203 | |--------------------| | 1/2| \-22736 + 1596 203 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1596 (2 n - 1) E(n - 1) 784 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1568 + 1596 c and in Maple format E(n) = 1596*(2*n-1)/n*E(n-1)-784*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1568+1596*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1596 (2 n - 1) B(n - 1) 784 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1596 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1596*(2*n-1)/n*B(n-1)-784*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1596 and 1596 (2 n - 1) A(n - 1) 784 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1568 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1596*(2*n-1)/n*A(n-1)-784*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1568 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3299801837207694867546977442935769389283931848031166333146738623359339170019\ 803006495212920842800077482056975609719529353743429467067697530043959927\ 979174447795726626737171196108086719396545770862452572716997089300004468\ 561607461302774441315767939994989457736995661543182325065827907780011806\ 054046522443731618475324308210091938773043906965387582051231743073959867\ 451718592815743606165590305912333009527958434124967634450081312068156976\ / 5648093608588178779267949833850165578380286833908008527643 / 335838225\ / 089246190977620710437458754059142586680007183198809325850494312620934474\ 042276551586286322291112867802540992483667952054284760913002793086800433\ 273615707520690029821861879282912298955146066138134246247842477983283466\ 791630546530098909684858445072483766516179664961477907136093648535936512\ 919257902253297452242875058164430712128713163852933701483746477544921059\ 755259407597321776509602894498949030257908731310335469412609432275515058\ 97757530186820954266360542915356258600803883690922000 and its differene from c is 0.27501279583860499027897426273330195933329522571393243629018436151795622613\ 647199163640168835216885643574744449897425484511133055253740810726906824\ 308647804148271407649887624554223827818723838718888827903632778707357502\ 046826527045931056439605139269955701740647943358518433613310210781272350\ 638005895859959395401208366663042368699628064629288736231666844519392436\ 799249568816801625117925910767223843013498860566564941433104922893003123\ 114885924218780409189630849364723940759230528531880851615468809983467507\ 382419592724345206015584653697853203058441265772358220092804738904754427\ 145585503327771026349390941997304082882005466904482697324309295134757294\ 533588392019844167094270448359505309930766416607469514610066489552646225\ 543740688908903799342527071283484828668788493340367361487952025598752804\ 835327933107437854199310366492202437373685395705198168575645712427450510\ 392097266735491940628966003616683446678185841529960895315046158332589124\ 404706845649120393065119409220755826302778093279371584358085915996902568\ 181657665916504455148825541226821259598845395638828957303686888225705364\ 711151711899524731577532565465655744938349877659516608175678624277926306\ 978785693402567816671739313925190166385507593233022679706221638857768258\ 316008090027824686021311511150789590653391748959041427885461036185203812\ 878031611318546781183540843848438858553887911688128896752770084086270070\ 943115149425645407043085742799642283870885399107296734976009665463260287\ 444750058480758678365591171315679007330739899630078883210167300612601115\ 295914762084621723946391902626060049331029402361232589702165009371659242\ 204280496363419800416464195628272030020862909896088183857854458954956852\ 563237056620147474682502997223678217398825061120352642419308855709137266\ 032642584483469107085166056099907255428567532762734553616133033463105262\ 272846040560807119059299328427669730139586217393687209842043752394473645\ 920005184314403727342591988835782812692647194158699666246214050805292279\ 672622273799835848559847668704028164340261899907427880570306401278048606\ 966118922155307421228466427068228228348038769527001410780723991373814101\ 674750995468476515837957627437255674048679555776257021600243166821625034\ 291149375839022409052206903530751221297552739504549134787771663838931287\ 687129888049875358468583789190985114918475085139263347960660384686095618\ 115890933029822583045571403821553130281837982104057496191749389123941342\ 262307703562742632647929809220700364755938365718746139977829501353725702\ 578447066854953161063414228345159002722732284072990202871588808538991954\ 148279925272052617923736137570760987764787530625632490736973712815531848\ 750270021007055437201272422217449881299212095961601891330820184170087701\ 342666455122853638505577832595021922026612058203068009720659560023387508\ 848087799601822889278608785684911921783424241870498267876819300192453047\ 029115836937186034970612076920532721349428265824477418719645739155178824\ 226786405768949237155588007969278968470643841651802572560181462050969522\ 791329772175398813096348422176156429312121525231100203379477194648990448\ 648532253806404695098557423916837965016154183647220581393182895582358550\ 523327191541693527360646418766817507683727801333652669606644985609221236\ 169780958159962196258318228414477658691861192983457981462281621681333632\ 815106960687540076212531044259690004363636036834768447117050598423067728\ 848084551613332541850960822948056629105630196452050609431023391499195542\ 256750915737437413389980052870035538690125946770108428163966778709827002\ 977075618643819150652869243397741741442923857525866549139669613173806920\ 806745098762105212843975982067269559345900761693446798777855481582190717\ 956720907768762969787929263204377484179317116653541806636266156479653037\ 260998979421049010745467848055657492584887379533706640369698845618813683\ 173590829257768479336334866826927603353885492759399057263656206213100453\ 616775003304098091951840586968568054426999139626344340854564847821334788\ 096857691044893417997358512971042560453663802778790777440721244207553028\ 733165603007091086792816752442662654469123477529148779948367798124716530\ 269615598851601221425571120938800479028357197783641922205714644190193799\ 559878597006665836107028748615176856417907970153606817202429826653249312\ 028326378171431527383919177819605241668654085890938120541103402753195747\ 967514148199234115962351327930204987895707501665999174852515938576473375\ 902966303562134908794226827074253348773227613114314193167554203794871562\ 903024261565956437271632691038406944491247533618133035059804282079634131\ 941686702086802843049219532848389239499922468110273535262921295841967876\ 686370433720142834473873037230725193072198275466410962166040982450133711\ 908088470765720123788362238538309868984230128990919101623382045944734911\ 762646614940622452685349234985703189088875952254596506511182116908054313\ 083137585941168663540547994462952278395830959863407679860350102147352122\ 625125169619518693927560377514927182674566390943060073615851364754126211\ 278122152548094947462547547804328707254882407880806620789577611203828608\ 867732941544500568396397115328433630994355363000549195770386805339881204\ 440235299376962896441690638430092535609443198210924373606013037933712792\ 784290726315049520975188295282164762786106500946861082207256800884048521\ 667827101434297393150825517535998777185121713448681927896731538014674933\ 127313220363168783032873323817316995824217895351916297116194837128599472\ 915707809031537100444582172552653265705420405004053307420106468380380512\ 433628567643338452326924501396202379588407302976787737036941082054087462\ 580389098840014070607894957062558947032934315626886803179056389477799700\ 252187892825788774142647271710546245338738473765285062665039982614046047\ 571794877552419300352299266976064983972263732445658073377677880605099971\ 949041096726066578670695782611631860814627499050360350429931594272409942\ 680256291996563003635012451773556952783421924922585620364984000107779663\ 459190404963853639153022047650945108756099111556367788812233501900477652\ 671744494307591781646619234660124596526800926128345897588775219561789801\ 697044367091452682545499586498541167400693985389833067574174332143861643\ 776632546609032232240092652134444857462707916720525550078777623912013195\ -822 53194160431032928231576853874001622484907783482634554327 10 The smallest empirical delta from, 100, to , 200, is 0.6618771044 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 28 28 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1596 + 112 203 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |203 (3 + 2 2 ) | |--------------------| | 1/2| \-22736 + 1596 203 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |203 (3 + 2 2 ) | |--------------------| | 1/2| \-22736 + 1596 203 / B(n) d(n) But , B1(n) = ---------, hence n 28 1/2 n B1(n), is of the order , (28 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 203 ln(1596 + 112 203 ) + ln(--------------------) 1/2 -22736 + 1596 203 where delta equals, ------------------------------------------------ - 1 1/2 ln(1596 + 112 203 ) + 1 - ln(28) That in floating-point is, 0.651332382 It follows that an irrationality measure for c is 1/2 2 ln(57 + 4 203 ) --------------------- 1/2 ln(57 + 4 203 ) - 1 that equals, approximately 2.535314423 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 29, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 29 that happens to be equal to, 29 ln(2) + 29 ln(3) + 29 ln(5) - 29 ln(29), alias, 0.983144998594759098 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(59 + 2 870 ) ---------------------, that equals, 2.5304177615134250245 1/2 ln(59 + 2 870 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 29 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 29 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 870 | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1711 (2 n - 1) E(n - 1) 841 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1682 + 1711 c and in Maple format E(n) = 1711*(2*n-1)/n*E(n-1)-841*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1682+1711*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1711 (2 n - 1) B(n - 1) 841 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1711 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1711*(2*n-1)/n*B(n-1)-841*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1711 and 1711 (2 n - 1) A(n - 1) 841 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1682 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1711*(2*n-1)/n*A(n-1)-841*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1682 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 8245262175684343875895784147973501384872591296637703392904312689076170188309\ 497031565682345404874040923938120766127677803385492118044680142827976391\ 027705504905736826455210746185505320645700357322043621964068932870768180\ 993402455048665404517441340975420477326937815564193099292814226436460941\ 672862570495522829450901392332493875790247377756628708700892693578720022\ 869598132160379082176337014044296105699623281130330346430420281689146344\ / 13800150543490778058717219946819784941694183824310025812059 / 83866186\ / 447264273523546651465263640076179297581616391277804259989991847574073631\ 597564965717291745488391439136577562284815239853352275300500428815080582\ 182404126651479259187002446589775891119292400543948706463008622149345042\ 620896335948009730635486933807419990407809683541180238988286948248299369\ 277578407048414306915987544839081291944736705807266587490325687275451249\ 968275521450453346423645269139757673187084611643999555619848326466560924\ 0819297571658595398546023583551596001893537288818824000 and its differene from c is 0.28045295756236250314099420714269258874069573328073735847522954368070273349\ 581589241836265796150556944317410069170981645447327112167021860241056829\ 524293253946787827936681618177796268470603692193807929633089523564544369\ 830545291510555242296120975152996704678309300136151256319953915905183068\ 502621575677720838150203808154294756668752156369495686662664957091670200\ 053424722476658128887939042486456102900102188534292977760108365226878362\ 576038435516982827425321813100608838317658782122424470350046708206307462\ 005909250834461743583881646121717767199062329858652578603773827005221798\ 480179984270417836920685194051952711749369237870025232346993633761450343\ 545809783669038224212884048206687055743689761201897693678987923803679526\ 663008257138643262663348041452231546498073170773430542444385919790721654\ 240438420781978907468725381181407726712320135731499726792683348951795760\ 322439126748446789225353581672881176192676053387466115270648692430893546\ 792708820827616090861204049578061977383700018999333576022478781200052781\ 050076996685726658785308646158252125141465527473963610980686613479974185\ 836935326133759062893851992026102765417085880384315194311324221681048996\ 334403472031943885357194778959666340709201733559266286939308060024408427\ 287332503202658691610029653016783240425489236671105763870952820131065576\ 682796612653230948805381776975132287116643379342105272803812414941000660\ 593155998380660598581170337674625166883844510401388514061563302554217484\ 910024946643002156801225130458121304225180148596160952048348245527640896\ 829961335494158738916065817368630186433728910877551601935254919788383621\ 133868161582352116739537818346868779619349400998494464554378236222486632\ 157794139748687826732992099564604891890397778731113921674440545281739311\ 853350076583558752695927781861425275485064280700286538663702803784290241\ 101583632439072283356026981064161252118732036463361081108697330710888350\ 070295664617083267444725624075085150939725246766274789628335926496226777\ 167954821142115169817192265823823711972758949867235294154632612267464410\ 222814298402867704956123950959080038642751665888325422684744727369088774\ 965761678498454973011319533712491426748440332318161116165413554975831955\ 999075501094275243736653265847270619497624614746956739777115919939051933\ 928666983966494142804554100258913424951381574420740697855054542435653289\ 027602865746626882363406296691777797041270305964228840553563426484392457\ 295893864311061789048969561501216048752131110216156050293489325995771258\ 872912067277195726437520223544028090838243209666849954363744069304836294\ 383558898942690971263708263435356461794318271833115719475225623076403054\ 154450399230144829714459871354925006654327783497442935074116638355468498\ 099048599087358587437410992582130745312641046713206099482591480269703980\ 294996972822788629466318606552673700642512389212471582956404796456060795\ 295730362335118592094652295338494401928732010141160332729094475907170302\ 838703730012879009184774631371652471345948383632113519425096705932028363\ 748697440489166146845555036428182242052587296595229275700221138504303635\ 515523882277351674494026070311919160118863712623635010442490056684655436\ 976962827847637085520416544880546896550295884252396018571593796930810103\ 219174352044475729832387404416299252462823952503835754513861051407605380\ 493619249986551711203499438926788219905007788500082068799459112348406645\ 561186219388323788928515728664330703568164361193047039828924456854012379\ 043141599664123104809370870943809296703801763065263333663066070582400087\ 218558479927066554821305659572616780120010222572577580855080478776650113\ 366466175322597926996376007609718025012993964537188008815548955116742666\ 673589565153120619798915125929183639272231731346119683502092169320175510\ 205151410598942031066639202503227581380678201383842824746993038472048092\ 271187379782471013143063033243339944385577775637309672254999474238308684\ 911166231363372484855666817856880906206402165369444360319477723472783608\ 400039836510559606605964615768859580915260878162527212371780140347798371\ 195657761449078365249706391137769290542709107760854965478527458804546025\ 425264334186075847610993198301375721867394392294739877819140418738928515\ 671498897144013682035340623108688299087185629362268912036486776807063009\ 411466321048195499924271116083263060018923970307012956594674442474599349\ 650550180990988728886859071478272851339404226153959856819043618340059942\ 038623121746347084007708276192783649419563138681625650623838837827924712\ 592495555927678515756452759395294034005233441747339031835568831338251796\ 314116320222354810219166090228052374241641790225839388723962331199662764\ 676085207760647582239787828420706481503941632174240211310861701231395698\ 328561821450908447176908236755563675276747871786335647556430844400296663\ 879585172273641464090185833020131530722474576144161375574674823711606380\ 829553114311183988511633316815641194520008236421926352356875703842342759\ 802452848358852690448796761317812287605282720793597523858321290416948098\ 483233870795840343503018580663261978489156790781506698974738204114281025\ 606989713929768907235695824052401492811258883112873716955826891069600779\ 435768942361904343259956656091250182293400637749423931699250577519606405\ 953442917706267342190100017512972657134059223373577291257747999133152352\ 918527220191024057182667969380457439597085188414554740122415171646012063\ 149280880387176045845623534212595148446066234917304970265955166073717208\ 938933043438619527434657091012917252000044174102317242729567472804760765\ 503309530296609737494916855339071058238910244536370601891377538524233857\ 925582055865545603292771113481346149495008377617361368266115443616410373\ 235147132395489879519796859471534328081996550762812092025930563135567370\ 330061574732228752011990973028940475155283157163187011857086383022476205\ 108019364816328624824642524193559078585568327613518951320536872596761933\ 648320915645953973462920645921755185380728524610155311032024265155386805\ 171247560840089599098301556497833375365337059937631691433249675196437976\ 243443688469600263713095947832109470657780501945810970137634520799587238\ 968809023050011005010469494967711405717395062234765002018455958349498390\ 717624786045125942682883452155100602377628854682980498621775595525274749\ -828 04010926356413181384379959157534604348084567416037 10 The smallest empirical delta from, 100, to , 200, is 0.6648164300 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 29 29 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1711 + 58 870 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 870 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 870 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ B(n) d(n) But , B1(n) = ---------, hence n 29 1/2 n B1(n), is of the order , (29 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 870 ln(1711 + 58 870 ) + ln(- --------------------------------) 1/2 1/2 29 (-29 + 870 ) (-30 + 870 ) ------------------------------------------------------------- - 1 1/2 ln(1711 + 58 870 ) + 1 - ln(29) That in floating-point is, 0.653416358 It follows that an irrationality measure for c is 1/2 2 ln(59 + 2 870 ) --------------------- 1/2 ln(59 + 2 870 ) - 1 that equals, approximately 2.530417762 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 30, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 30 that happens to be equal to, 30 ln(31) - 30 ln(2) - 30 ln(3) - 30 ln(5), alias, 0.983694684689726117 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(61 + 2 930 ) ---------------------, that equals, 2.5257687337860714086 1/2 ln(61 + 2 930 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 30 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 30 ), it is readily seen that C E(n) <= ------------------------------------- / 1/2 \n | 930 | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1830 (2 n - 1) E(n - 1) 900 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1800 + 1830 c and in Maple format E(n) = 1830*(2*n-1)/n*E(n-1)-900*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1800+1830*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1830 (2 n - 1) B(n - 1) 900 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1830 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1830*(2*n-1)/n*B(n-1)-900*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1830 and 1830 (2 n - 1) A(n - 1) 900 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1800 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1830*(2*n-1)/n*A(n-1)-900*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1800 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2775981958853077733946585019571103049974956311750692592692671160728239423943\ 034150205887847158854480283784455975987407702609859849224135718006803838\ 512521330687606336923011852711787353994488645719432067056021712699015637\ 473000997911474039083673698665460652953038260788113573809907871777948197\ 210883535097368019567164301273974731087735407547465102371216664147804119\ 644339056889172979919400312181193661152438821652678067521104101489744875\ / 690989427425951261087356501930067975951088836789583655404478877 / 2821\ / 995485040837884769331580555781905533726323237069312509353360887602846146\ 963813277534730575244614819413027980650653294306106259055516825664827782\ 977112071500269479060797026242926123463773138103382246799282287196271540\ 260361192685826394514423820097378387019533917131849183992634400877770117\ 693071975968589742282519653228449700801113926590358870987611842334931066\ 820772289007975330416418321426672651224213747157280607740124926696498924\ 081723919500463986529716377200406073224753067362460398626450400 and its differene from c is 0.45307410299757748100453271340266589100638307647843084791760912369668838829\ 306551487068894954268068356839886321165955822691632332871279700275024593\ 580860762445723449973066563323014543356683453430376589026305659306195151\ 832773519258985577305031150600907753333396755240273915999733992889171629\ 571936403196397916338194720291898477717582782927172375311055273960327115\ 305342293333632649681467995727182697175665293323115622727524956641542638\ 143037311769070353940986853878440500782905438016484391855695090842000296\ 449147664057844385740373815648094270351116396573130345441231909431880420\ 125311294866281273958240906095620158730985073866011495015683462939072602\ 859176649072126706154879195267779363566299123161185013801514193079334705\ 612253391841879443538924020232559936206750791788747125558787805182178993\ 091150292665964379563740251768202419850310327951505576605837173468848301\ 450239345207413731302781185781771151450879724106432305117033034777302351\ 680513065462180773487830003919483862192473168904605271483520673489040661\ 114773634994554396957459479769933564963464450759728580541757432195471820\ 331932751077956600640048868374050349664777087477125759622361150238728302\ 634842068661886543057187493819020408699465571107004877402355873119213494\ 020722984196089998012555401173045494235104684898362562460281087242760155\ 459981656698752331699949607685271826467515224871733391243281361459140505\ 547391425281438486122226437492650160969213269175802403941596589882935782\ 706591696135238606358643323572641987695452891954944683127908265653485021\ 727656805719083122291823946867174717173884925828121607841752388433234586\ 454164407380174778688179245517648696892109958030625016671837022754754236\ 719984913625222200830650734379892496337907301445747226921520273358716279\ 099061514252415293423011617927555515904537075459398719146791436389358921\ 483276047096385245216440482136509246361794488002483494840759059076868756\ 990281532367204341505609086758007163872306868931605695476057047216375061\ 334273551970623788276477650387639814186327951555020348590597479429481467\ 012979091796141958200483562931290858385599779180749253214727086143989567\ 053476754886929527582002753697179255779914633636905376793690006825684403\ 946819493352484599633100339926078561389537188389697822175921005777454289\ 083300452875039354947552211467964338499499917380979415153045877038910551\ 867381691682542441625554459195516672371785491067690909398154053152186808\ 996160824012969191613496526838326730744764141490794212421451298846122252\ 662558265685948093876854613945819656963349387591391858113252829174982868\ 835772284550461022954630641165170727245770177671821036240442107133497168\ 284711284176397694576903060276112003560148355311131628771190422069792717\ 476439659925487686799650977087294749612721309920511437159540829439629394\ 721370288336642416836264371883491849532625739485673560687989597566932005\ 722545743481106682449672812368265526999886349792932678086243983900926006\ 719739078467649812431895095717962167758740392143845805011888209999689709\ 183199234745788394402815644167928504525099432484823089842987214185254306\ 843612239309586656378795154669314639652835904938909913215769380034563036\ 806488111691584948940134265495391095871677012306893459391660620690082520\ 573340518008140331110502454536526475002764273011106200329393833622127077\ 510605589830178136802351960593024051797568318474758654164463334696095972\ 996219283856029083665394584155354218042916352626993638092257032992664280\ 308656346184779481485741725640698653037369158234143471397444927018845792\ 876983605025778690063135694727314939921656458042936409218492396811028885\ 759863459007788925429449848967801726364537704170867211351868626322684138\ 344164394170701369783379673542946805477094787869957643615287559905796491\ 566148717648673842848679821575669578029085305033265815395739225219518666\ 504562997953696188252684343359702364032445636995555573221574002802151574\ 758269431083635132789547127588002749176206261990158891316422960802213618\ 986052424269327573775169513769595218919860054840157960907943199982409335\ 477195045131361187616835948776718928852394613545326506958168696933987011\ 738301810588442760145200886031150075870836649478872568059569465360084355\ 225167860388058742099917931817808559082730360830713985369481921337934564\ 530195397668870276180857606382102882737325462956282133342173422050610574\ 994936349832873820948281798349899797267946310197220502100658805756731557\ 579815632805948054699633439629477954601277848334217261664469364794717567\ 451602582563826437690035824916755045359843332076551590946328346763719343\ 076760129973600126454791893177414853962023303787714246781073713773152322\ 133925420316691549564405666058471780120766515663165183067796370374805010\ 914523855238872613192591335057410423955461360043659016310219208862872346\ 156707444008645055340819823335862783887252219847745336442531089803958807\ 968824685032160862124095323541313700887666943352297492024920184276307584\ 959931489858333618846878840724931349972478652333103743701331279569022642\ 589429730593977285504638644118316636024084251849325181915234153762648381\ 966236303586881131697994773859305700804549741242186040917699995560331960\ 831842417635665704741082774333584325799401943150607574023093161597446015\ 366857478648977353951044441973125556780593273432207719919728960642889357\ 040626578246100946218184553706555190066311125314159653001462971466280859\ 002054502314455513778520992583675058727594004819341585543488868864705398\ 662478591766485731997348268314838327519619863903324690563029977234899317\ 876275660446173547177866534217291840714949574977152964263060919583372877\ 117751632783040260455556124439141448594831423016070445003453986111493272\ 932532024173246645661082158986699158307715911907147147498743238340530576\ 594950364172033750717537509679020798774801502274025572415315844448033606\ 946455708914641399222368241517725806584386035961607182019016395108997705\ 000846484814536315539049211864919646404976124401341683789389803054494644\ 010174301874246424423868373594329741920447555545270145689761599028376850\ 658825260878946421258024739075703061501590077918812423411245059444077824\ 481446809885483209755906986804306467395896786461269023421226111676218067\ 900581419622899107611883665455423453361380282452795165828085837098394464\ -834 75667915561520335427842414541427770247305777 10 The smallest empirical delta from, 100, to , 200, is 0.6655791426 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 30 30 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1830 + 60 930 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ------------------------------------- | B(n) | / 1/2 1/2 \n | 930 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 930 (3 + 2 2 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ B(n) d(n) But , B1(n) = ---------, hence n 30 1/2 n B1(n), is of the order , (30 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 930 ln(1830 + 60 930 ) + ln(- --------------------------------) 1/2 1/2 30 (-30 + 930 ) (-31 + 930 ) ------------------------------------------------------------- - 1 1/2 ln(1830 + 60 930 ) + 1 - ln(30) That in floating-point is, 0.655407322 It follows that an irrationality measure for c is 1/2 2 ln(61 + 2 930 ) --------------------- 1/2 ln(61 + 2 930 ) - 1 that equals, approximately 2.525768734 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 31, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 31 that happens to be equal to, 155 ln(2) - 31 ln(31), alias, 0.98420964775198934 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(63 + 8 62 ) --------------------, that equals, 2.5213466815736988672 1/2 ln(63 + 8 62 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 31 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 31 ), it is readily seen that C E(n) <= ---------------------- / 1/2 \n | 62 | |-------------------| | 1/2| \-15376 + 1953 62 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 1953 (2 n - 1) E(n - 1) 961 (n - 1) E(n - 2) E(n) = ----------------------- - -------------------- n n Subject to the initial conditions E(0) = c, E(1) = -1922 + 1953 c and in Maple format E(n) = 1953*(2*n-1)/n*E(n-1)-961*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -1922+1953*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 1953 (2 n - 1) B(n - 1) 961 (n - 1) B(n - 2) B(n) = ----------------------- - -------------------- n n subject to the intial conditions B(0) = 1, B(1) = 1953 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 1953*(2*n-1)/n*B(n-1)-961*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1953 and 1953 (2 n - 1) A(n - 1) 961 (n - 1) A(n - 2) A(n) = ----------------------- - -------------------- n n subject to the intial conditions A(0) = 0, A(1) = -1922 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 1953*(2*n-1)/n*A(n-1)-961*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1922 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3559093687664590066064548455076075463750504482357587532201569918841456315447\ 211683953035716673725144522045736907674391383561912320706505791327881418\ 561438319540717263262724935981965016920701767742137793984773310101306629\ 737718621336488526544970786760071548570191093631467762079902311069868093\ 761552225463800393707251644548557898488449091507365984288369191551396009\ 316507484563936876985757100518557276369603511514722169898576723031599248\ / 96241060038760477077799229422352849756859651434663147216936049781 / 36\ / 161946753863206481610764486474983408449291254266395631178622545492943854\ 040717325819641646484033147577367613247549986182614020030568203418962910\ 097505901291227467175008412419546065638526226896770283726051191425218293\ 528825658430102263819816265049628838342120688801323957155380033629099622\ 512360300812010717124403380069445050634406424433112325117500885671928700\ 730478619362059855435210963295327178468303785942016811716817171708016807\ 6566720079566985252979857302573495202409095939754025374801389288000 and its differene from c is 0.11255944521103776154350962052963658568549106045536814124727975044119839086\ 221674138607346538237786764633591709071037371090696366495010011994430306\ 192280495073644996986136650687622862506488165085349586026439812653129937\ 671614501416685891928136519114932269801266800502198695081384898504176027\ 468772613204301832477322559274264255012850785301342618132233422309931016\ 592366761810045789706099770177765016214093555000406591329931397249013595\ 902807536253925158337766100134699635071105567290816563992860703471648275\ 638073879289806047118018779467242001905167358859078707443958942869380003\ 480708882339672223741791445352320679966361732779099117793333323598682574\ 344937054191838337459762384181032798181365220565203842172271296517419099\ 039169411552108537643284706080975265479514241007761281905397539759798496\ 660049454523104786894738965030184778894518535293964515880354372270106874\ 896643764667690107174474730247676543969279853321595782151761432639569938\ 422245174321089667641804099122472411897239150478692107551639394637566653\ 993946183573218403788302461854313905319492817940461975111278113062588730\ 176337570498031084269093980163182409953407349051181000003386525527296706\ 582025651945486923603174685728608360810782673416064747238271125100663086\ 815146455912978012428734823409320099356070480362352836849804537465147430\ 615954606319373623130734175873402547145129235398291991683621964649816549\ 037574749997668104945903759067140145102435115618608555624868044705267663\ 189676795617927890210366117358896856936697578647315372847096971554715924\ 357439031871282485699269219259537357403047510672673206939590403633022984\ 480695723864886530671926487380576682041654380495403900601480630978749784\ 517658025356045078327882799535804509799262411219214739997992358480346567\ 666228151057973144839193820787280240745953247073290998985495814274159950\ 423377625985083329625171890487440926986456022471323969487852577634154345\ 758254431037470421003934384764594660536583587875798176258957732714042070\ 035528995312600341848991886498529428656892772016343114860437557672762851\ 109973348169636898581037837134634991093839091996861249956368465785644886\ 339473507866946713796006746538440795985399333714800089025759031213494421\ 377427748254950043028252167835237498985231866565097747744639324307574822\ 618599262667901426257704418554294768214884794017510864921458193508207687\ 965387269650577499926258786689468396175808087523951434210675553942651041\ 647016846137318206852202511686611261958669381129785931512720969648157692\ 759464706568432599485485740483076326283379076864436489628188934187553607\ 999906811210074680955116750843735939870791127079754955988889209420169607\ 590056981953877934179175433315007405826659145400569741398486603911258236\ 518469642093101205937922408276926106319210854354844422162123583478037086\ 201431645318381536542167716995925124483167306982507702624061774866110361\ 644065033740961876262268912409622904320803308640791779931940497124362891\ 827056513651113314289854540748932977501864568400873043896322251675986967\ 246359235596030436470093528880283626643673917404533023716549238604251750\ 800448041551646103919413518704897463868587734871863137120270669264890718\ 871815948026585507541529336003616219981451470706725027379949169445071896\ 179957219467077391361074126417213476500708952113041070699119185107758953\ 003667543682617811685321007952252263734727280242987493493695281724168493\ 690064462433072239472623724381968806180162497880240004866694487286055320\ 367347391779929665760807013347065272160993118186966208708816141327271124\ 445497957563776914920719768390320108580079536646003926207016762005958519\ 617911485254252454044725352079681526304028390718179503457128122698217303\ 408023843840446819448504925740466686285221310637744063219067979739586106\ 046232935186221082209147739085846243178873295304175521925534625251597536\ 445386300847154440136838162367066699372726882321700468908329661820220519\ 037614686539551679735803572189283484039215739237318899505927736816635114\ 623371567462716752668276762994822175835587129984630958556440389691698403\ 298880491554751189657028133760843174005836457083990600579773552140710905\ 075370712940393828099155823259529484240832628297819074090190935510655378\ 528199955306751439446245401064881661670031272198420386660068144405684479\ 299483475445277575335193123535982646388621888150059254015747667158433684\ 882304787429238227827051612883300966669286142427314785941290145693856013\ 261919949247507504144812189882730599220009253960188100351420922715059297\ 474754252888269713345470836393054607647217705609549106218040296710169346\ 073219583879171407019893371377162632465259375157334207473066852915868985\ 808844604119619505764340682514736911426107204591718252754470268712935180\ 151159092600507530877012924872913307288026169928188490563855264296213065\ 456448191223919133527500815620643582750548303855799111775976690824290692\ 493365443367158356249340388686377154253135327376881491336717499898742300\ 966401086351370779346541660339328547533840397234497474831792112019298565\ 618809770030638447237730840595889194054601556969455672046695029725107628\ 145258549863669761880681634610502385650298223296450068100885139762096837\ 770713408929918911669637867869382850209369808929369354280562989162994924\ 305529842608780024217094268386962910477212350006494422655759308963887682\ 793125800325961069265745511759311743623161643001259737687639030019737054\ 641592357398025730236381351305381752605324763125041495746591433365815406\ 643677535669171178340247525576800342706832615427725986812474519525106535\ 046532060946887724353536719783556289685513733473850937462129454823065112\ 097221583006724622574980854554918564128254534004703660975019246711725602\ 911883552618811118338374703187475060478468357981385244250690100390478036\ 443358991983709172887073247632050897429435137877384293078611350213796634\ 175270934548006117876404800486665606410043088299222164124797970530909909\ 145483505730998512449870094561962506304837691175244650374448509443248579\ 794255718006252310431240226636633842359376011085160938946142026221124312\ 515642450436357163516353559764411832414413809636784430155577032253950443\ 449153150751071479964832930432727957926109357493410575382487643412691797\ 568328364264510650936840482479351485313940872807597394430164056290740710\ -839 057870946952454177763932727300967807649 10 The smallest empirical delta from, 100, to , 200, is 0.6723439762 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 31 31 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1953 + 248 62 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |62 (3 + 2 2 ) | |-------------------| | 1/2| \-15376 + 1953 62 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |62 (3 + 2 2 ) | |-------------------| | 1/2| \-15376 + 1953 62 / B(n) d(n) But , B1(n) = ---------, hence n 31 1/2 n B1(n), is of the order , (31 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 62 ln(1953 + 248 62 ) + ln(-------------------) 1/2 -15376 + 1953 62 where delta equals, ---------------------------------------------- - 1 1/2 ln(1953 + 248 62 ) + 1 - ln(31) That in floating-point is, 0.657312193 It follows that an irrationality measure for c is 1/2 2 ln(63 + 8 62 ) -------------------- 1/2 ln(63 + 8 62 ) - 1 that equals, approximately 2.521347102 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 32, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 32 that happens to be equal to, 32 ln(3) + 32 ln(11) - 160 ln(2), alias, 0.98469307733611803 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(65 + 8 66 ) --------------------, that equals, 2.5171332351235844357 1/2 ln(65 + 8 66 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 32 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 32 ), it is readily seen that C E(n) <= ---------------------- / 1/2 \n | 66 | |-------------------| | 1/2| \-16896 + 2080 66 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2080 (2 n - 1) E(n - 1) 1024 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2048 + 2080 c and in Maple format E(n) = 2080*(2*n-1)/n*E(n-1)-1024*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2048+2080*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2080 (2 n - 1) B(n - 1) 1024 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2080 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2080*(2*n-1)/n*B(n-1)-1024*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2080 and 2080 (2 n - 1) A(n - 1) 1024 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2048 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2080*(2*n-1)/n*A(n-1)-1024*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2048 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2061004822245665129172555352047596009390319368966064998034820696181764791448\ 281277034122745121422019782298638817997415518889731468539632147935423725\ 683816793727649043223131092094354355461182294601817095538118752422876192\ 968224069499440669310730740599531508031369246615517621808817274632400488\ 325325460201065006480656167930406052846146383483639206046012493155694227\ 214625507459857724882923462268994190483190886695587068314396996075949242\ / 2285028540447704279828524850774007361443822255666769624457034296667 / / 209304286755146516408148963652854538896688531676303572274972548458055280\ 254166266195799617222367694005141362042228476339988149609342302768533299\ 743035792905725932043121841050085778208554290064457069038836791639010108\ 924168844365036716338414039937785031984127328488357037681982627684815782\ 496883793342945023504421840855031488812710468944886836879491628779975109\ 971441708446198003265574998956023980122222169847589994674371384638744664\ 74081608549071850021522205224590418620562891700608340976399438641609870 and its differene from c is 0.41861016093309936676106155071173312976343120635063566107266565117527510156\ 407076945056380212987792816285278305790230699855031426297651644701250016\ 686063548262864655045254297366406785249536416132507800939748873802417196\ 618158770774773197062364248959553572760811001092561227785489443566592281\ 733919239681544376012308132260483815142827138358494792081622030154512291\ 024133886727306533947142299856679491432022640665893508485296570936258463\ 907476764632630133968079785696091804455609477236852634788365163977966061\ 818348840591931343134818734215433576122208816959165839246827510803881374\ 948038009631647325084727398470892373312167118387944683957750148106606219\ 504435888702231161841734603131247600578419640032518226626586930244452889\ 906934681065618460784364718208701743302782255327815897539101605380778782\ 876536097981323198661311305582456514235890822335276012636953111869472767\ 071034198889145235819588876104141872240785591604424218133907837725482387\ 934837653713413706645242998894291517556238644930620771173772338682443780\ 180737642473858509533349807489378884731031916046488923182423474378237777\ 162822675853065398888676889981687254283266894880487665772998023044581548\ 036730275090414262730069622005111878058276759851960304129170278092668734\ 540098588713016156562367134833334916685149142112168203881501670889411848\ 843792463097782224005080431464572891107260812009767215047170814778403293\ 275458643068507244393865200826592391973238684232866396244302348071812187\ 820582314574215643887793691734977957335562178259688980946361901603720533\ 333893763803784975384374032262573204012797578594223804363439765742959123\ 575039456717333832602326480487775476152585035736206707244861843520713051\ 346155090313027914242992648065982067790317688156211811115702531938928016\ 995403438997470183574911226593744961331640585274450753008762442319530282\ 078797639012272549542477783182729805883643370197327600677266931421160564\ 291360863258068227705575013496374058907368146992648102539089752945678247\ 753658979609420123842311951079934725541404759790931509238400519479432139\ 722061352858525122617878701008993938137636713852110955728254273041725333\ 434554762540905584995099958799008525881954145489570640863419981040831924\ 606357597943846212364318714864744672276147094783168975198696949903065289\ 636226092265390852316208144922178990054130708168675747880064118303569277\ 143705496226209372997968920945516332349093656489139173989156654380287628\ 358037719628323109638943137097875432993991874152073233596758723280804264\ 606015371116135567670295194491236753330689471261400242703018045399641644\ 102559091274524686466001391626660160019310543995934357084625244706890825\ 867864919009538775443440818042540390713905521034893367860838552507816413\ 003106451253581949167883029785935851018704951081668789789091469253098718\ 164414398652604454731893075656217250538074844626930172016241540437188389\ 213031289447043111516553896780568305203911969265620836200563298232716473\ 466027896765419209352633340907165684317812087012425181488806697882556954\ 246246773922886405691537130270011621644726029180302191101544569991222268\ 869370244364931993014581687015449494280271986470743960633412562292311797\ 589502657687808594435205682070681082840870673834827836257191788652544927\ 192842765664294595188050097567821030357785650618244215807092986890519361\ 367352013934349583025568182199241334607153134569247386417297080190938133\ 282939173159145880090115272680186388589745178637030008711803667805490426\ 863581537822872535420481026082721012594785206549272769478080212455512130\ 582248473164676201055449723103763729434473136760477960890980787817116966\ 211809207750419904076079986805913350436552546848794894876596030969118481\ 880396846623823362007392306530246493077869336403539764280745990649331070\ 803235211824499671167925512953776894250461867786854819357399669848984961\ 428620124380894739815218171780894002624576254060161226999200048285501040\ 307387072207907614138359506827256411291717813482369408661304612285102762\ 400279716084786828198239843071737273997085420670304236340464342667607024\ 679809490624871672734204071560848860322732157901920663834616234276344027\ 025318475685437025827497178627365604361995317372496314890175447894014945\ 183169826541781819679816777469627104205850045734421553902717720557624998\ 180097720745463397237250242912874826207074564327001666910714689501593510\ 698960086483381960496998877666716841721486452355586490617503890816374541\ 362011643111558244375590045047166376061675806717132454162837302938250995\ 400750933862219232208500745573789744707559920558558348421460982524516620\ 048775769858532613901624072047973608728520195815889196876704208458577381\ 434279407949254094909285742556652875678569682591820096590721128980631039\ 933173979541934682205784969271039888291838765849394373187251755919954378\ 526040184054009907624267954081567317025332324904592577743398017929403157\ 655851026102926589095971232339496701534244272124826899678607868763829974\ 692856200282200560883020837209951814448372637222634714056643293346968550\ 693061567887751661614573247643941923910326047629198878410847393627739989\ 101708274294214582579722866121428318601265663724639575436681496014190279\ 465587408385416108143515123911822322910480024580868846988956725589369827\ 081792815418827629736298508776942081546793596564431699154322145638454669\ 996687511987558091331303021915727460079612930792200837125988961286469435\ 066154308477864749889165173757796306368973432987892200832553381838908380\ 488279118735491033090319848300592454154750611889106580162292728117326811\ 783558339825764373850796922823950184280992706027262653528038521544940620\ 781977020645148558440297611404935203989333155116613866722160029313929833\ 643910809963522321080385068864077660223200897173588604915976080292619065\ 968288095251625053976012621474419678911028271937506234085646512527429696\ 038528404928238711524369863749941862098930714414785811066433193075303807\ 848768827634137726660718900530258400934541159683485027574221160084088607\ 957565164714038011198241230894289919271189221490968532237174479710117579\ 618279794252070792736816174337173173801217995114840654637119350187698674\ 404234703764811159164755189758972763298453171913088743305127188445738735\ 966274426648286981271627662264888950178949440987668096953436404724905227\ -845 015928994025953865222713321583575 10 The smallest empirical delta from, 100, to , 200, is 0.6637932919 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 32 32 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2080 + 256 66 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |66 (3 + 2 2 ) | |-------------------| | 1/2| \-16896 + 2080 66 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |66 (3 + 2 2 ) | |-------------------| | 1/2| \-16896 + 2080 66 / B(n) d(n) But , B1(n) = ---------, hence n 32 1/2 n B1(n), is of the order , (32 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 66 ln(2080 + 256 66 ) + ln(-------------------) 1/2 -16896 + 2080 66 where delta equals, ---------------------------------------------- - 1 1/2 ln(2080 + 256 66 ) + 1 - 5 ln(2) That in floating-point is, 0.659138030 It follows that an irrationality measure for c is 1/2 2 ln(65 + 8 66 ) -------------------- 1/2 ln(65 + 8 66 ) - 1 that equals, approximately 2.517132914 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 33, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 33 that happens to be equal to, 33 ln(2) + 33 ln(17) - 33 ln(3) - 33 ln(11), alias, 0.985147783939478089 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(67 + 2 1122 ) ----------------------, that equals, 2.5131120520885463551 1/2 ln(67 + 2 1122 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 33 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 33 ), it is readily seen that C E(n) <= --------------------------------------- / 1/2 \n | 1122 | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2211 (2 n - 1) E(n - 1) 1089 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2178 + 2211 c and in Maple format E(n) = 2211*(2*n-1)/n*E(n-1)-1089*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2178+2211*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2211 (2 n - 1) B(n - 1) 1089 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2211 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2211*(2*n-1)/n*B(n-1)-1089*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2211 and 2211 (2 n - 1) A(n - 1) 1089 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2178 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2211*(2*n-1)/n*A(n-1)-1089*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2178 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1522460644933649417778121572366561225987793186954493167190022195028955285339\ 206456625683944980152244503018607039360382094544353724270143200096919443\ 247105138050026636774320313511851724757960077247196905173276431519232353\ 595585946881347658090009261532916794045043093396726801106269889980072370\ 640939485264681426936348562574741913546902805127583925044996016287613482\ 951231538727032894846889468102314585809572708418836597723980968297868413\ 209582796177112303485093862892895468290214675592358413660964278271678007 / / 1545413459537539591891483022391538370330458786225814957632420198093\ / 225368712355628335956978555975432750343768944349499900761016645346933011\ 223397343655069015127749980613397941888121218011193955557379291010496266\ 906504016376933832082038894499131404754249980219864345559327106671808068\ 593964592971316498298718036548163489743586938841153962802408944567810084\ 802015205528543685066685090105295108618185602581319335487682183481910963\ 831854856845171333156653419150534034969746501852861716409104285076958726\ 003848000 and its differene from c is 0.22741951257306107147230750113119182778368709021865191105188332270585149443\ 744987034490670508337499577046594610324297729482415798920990107107265780\ 638372394764051459273236069721363511713913366196285625471053842695657583\ 169791793600527535650540054914982558013711189370325624683639886175201208\ 552811254645653718360813369076062024734037118147746008253868532968571175\ 106235069178811678124504306836012218589984895613218098467373855013420073\ 474711304167732882336712860610904802393505099201106159810547890662252819\ 588343962255663246100779464480579353831497239880343190260684933947604014\ 680810522456356811298110800357583829685581316231494746320246858241919586\ 357367293457972144161749722824021593231437007929453518542328587454604583\ 856199642812516139963860761677883862941489509468164424655914398716224919\ 257786335224112851644265719424332421096324569267220335303524225014273511\ 069972259572418992042558931519233619626880582605268412764900913560430911\ 607396661284920321780633417367773919419535531152247021509394962330822525\ 429537711103798364086204459704818523650255588622943642169778003356949986\ 090704742216489517042789325901334864634466849515993567823438716735058155\ 203240887569664626042020368258106356825923222500382661338453942391884299\ 621376818695422646689105295498008196612623628172506889438117973285144032\ 721648479357971933770535466678343426629733651256297769436651968967837957\ 287751773325007326495550493601886883154297978923996788564485493727663754\ 894206046029986864714280341371944339144268432180933208748923404066088088\ 530316750985102433998951955733317224800644080404121478712152303264634454\ 642617429028680858994370740516152219162663400246806398724687008228972081\ 407172790881554595826834269081800124240270264093354295616277524574042768\ 220288115154774811252558824505170142046906985464654468262265401586926022\ 099078363606106498277435511044296415583750383401851139196617790324724808\ 080061906061721761349201560850378795839548808929952302909575405867457057\ 987533292969736050150469614081818029654007360937381478560716732063603739\ 797615300400129373052557629963918944395999183807018311077706810736338726\ 790561620503582915847209745011380862841245555561591190190717120701088604\ 811273498574625642777175402107811023091636417279825596429356602868284793\ 336192759517485026850845315221720310609750611679986577102381295616394585\ 887052235093114947311433694261190827281341995662401574833356178598658153\ 927145324308763026812287281648754129448215567791782413686603986142413825\ 378279985553739902795896324177022323481380025149104518659959352896033498\ 874077168271091203408179969653743322969451085591705360998229701205617413\ 244423260616307531733566602643930265214637904923522060299933244967461602\ 940111263066234658654048206947314396269155395640267514572023175687311868\ 759601025005486922554378838248361055489716151478845758999052979220802196\ 237030412554229968855670040089248769844400950990616582299602929674598963\ 467602193098201052088246013990000127687818781362080968214660476418770915\ 667894502095043133625475956868693939291184258804445258545395690412635467\ 585493804916644450783257335108808955494198730112782232846657486988128037\ 201805630644293837565471505356082327000742366390583656748633998537865873\ 089249663508093378895260103874059987955219356685014809560556660722905678\ 530423588336524038031840789128614373826106937101410105421606849171439679\ 681959095414230353805722996342382448333368818482281982389610868564199030\ 563172195407856269886697225808426082563588689383663174804383087184468374\ 565169095276052606606321538082885652948152697407152879376603806768249046\ 161757785143756865057494124431979921242842371280715315860228284942853679\ 797772759691890978928380142091734897677622152753197871763077405595249403\ 094314813576756635684087418412216247496138321377248123354841083409799383\ 602907674631094549422889541569899989308404458630726386330219957467285202\ 709870507622122912790462358175916541902112525693948763894941767371612580\ 086042202215525309497972609271170133408195028446170488086361194301589973\ 549466381042638565775011771512680969294268927343615264044101058900782440\ 242781266754026681320232820631547997152701531641413304225614357557356863\ 712225507630129359637356474347191410344082414315207849451156650862700217\ 052979481235497923829574493530578773339590023679626038103893745536284370\ 125381447558682256113735883760419997919841360950309931776337338986123470\ 483184923485680467558297206713344107122561612568267373095844237233135186\ 582915019781768354408621637158483218461034767669054986076348277751018907\ 759130524722404191240764603991746338407174313070275141778416736705714561\ 667424087333990873728551464127228386895639320837995648430840009063897141\ 687182108579436917307974501663363606776634769054369009382752942972782393\ 061448441710091595794460354779935324436540360532425117137926531354174074\ 614729037545681071424752828003188868736293787416544696000772282199308836\ 065607316445436626306523161196400739818947040879434786759185174308496951\ 410483232706353009666085121896000264627058892156576684465287097844672958\ 770345719080801907428684269436356155695503873659750968384575382575924642\ 470649101429632779184072854215357851273478581896036772640203814233056105\ 014637262845751126327541665361941544583624736692161633617175383501099587\ 963046669548412830727334688709901855904588174103144414633658725442322717\ 573126520482312437045963809500954217297022156526317788302710577282666763\ 358641237770301802549457790106382757882006667641659621147277494678454054\ 889393434544468838596559959291206186338723550713483431202317869149244446\ 347383542981450262718278213788258041116765253734982124393544295402214241\ 704955502817799459470037252982405430126742823631578045051450220153890882\ 779809698705852916372082551643689831339358068241537058551680475233379490\ 424756203868938213414396329597850456370833966040526684215958897439549728\ 972814720433627328597076004549387941540959407265942212211973718545712807\ 788691836734253111988118483590113131946662308570897491814500440436245264\ 439041729791305373437655726912003218272648222438692825633072166346894334\ 456391204999229818053027800406322791556000810520817300754537968187125540\ 700092902009596965021194938829681797453930214549723318173306016329956780\ -850 6546489665714892027702317353 10 The smallest empirical delta from, 100, to , 200, is 0.6752164331 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 33 33 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2211 + 66 1122 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------------------------- | B(n) | / 1/2 1/2 \n | 1122 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1122 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ B(n) d(n) But , B1(n) = ---------, hence n 33 1/2 n B1(n), is of the order , (33 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 1122 ln(2211 + 66 1122 ) + ln(- ----------------------------------) 1/2 1/2 33 (-33 + 1122 ) (-34 + 1122 ) ---------------------------------------------------------------- - 1 1/2 ln(2211 + 66 1122 ) + 1 - ln(33) That in floating-point is, 0.660889587 It follows that an irrationality measure for c is 1/2 2 ln(67 + 2 1122 ) ---------------------- 1/2 ln(67 + 2 1122 ) - 1 that equals, approximately 2.513112053 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 34, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 34 that happens to be equal to, 34 ln(5) + 34 ln(7) - 34 ln(2) - 34 ln(17), alias, 0.985576253690577863 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(69 + 2 1190 ) ----------------------, that equals, 2.5092685383031663785 1/2 ln(69 + 2 1190 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 34 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 34 ), it is readily seen that C E(n) <= --------------------------------------- / 1/2 \n | 1190 | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2346 (2 n - 1) E(n - 1) 1156 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2312 + 2346 c and in Maple format E(n) = 2346*(2*n-1)/n*E(n-1)-1156*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2312+2346*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2346 (2 n - 1) B(n - 1) 1156 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2346 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2346*(2*n-1)/n*B(n-1)-1156*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2346 and 2346 (2 n - 1) A(n - 1) 1156 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2312 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2346*(2*n-1)/n*A(n-1)-1156*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2312 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3158790933979812775615879723911898765007778418789263070404337425749096481284\ 000580264024144000034698333303342093950922998187105016660667531754227017\ 552711652882006487529863807054388605376923064573320630343204433960681283\ 868967511012380321139992962533775711073216130370920533854690859613915252\ 903205096577832111278731558288464126045178650215376898543933957176570097\ 273041351463999778817054001509004953739356543650160683768458192557125924\ 611919409819282997486596130268518849122243100740037951838014218990279404\ / 03 / 32050193195619714148760340642435815260056237065188995504951895626\ / 862619916135095644770430298668745172822012939886301624408682269471828446\ 228306972971930567628734123517530925422246785779041814653735866206974767\ 836410478967844103685202328585062923771312113724497354976101706542044276\ 014717499689645425423197387847121826473678555061668889041683907922261865\ 291234795403126290089365376673015615678431756670515574988808547029221780\ 358427983528348776178000085463438390862129855137175446481244419630033620\ 2088634348000 and its differene from c is 0.17650174779556073466549235183906454803742410854693351361946074118294572752\ 053390146240392452960055256960965758549904351773548515014232862810135044\ 934694275329772840869060544635817948715248879081889206290684543522025791\ 258927568121254330305705239999159066747281711159443468707132473077604582\ 769481670393899376776558248877674341716659387762474883727828895591523269\ 542597920815505547103881267760264117191839517763699773416495055571967118\ 588776904758061324050071268530014659175217167919253426909920150690134340\ 176550992344510707509750927578201925879755446029074162211289319471033070\ 830255298766256545377798423065356746496569824315198326405178957103596638\ 652162524303247768903797948153877156510583796321747088376951180280058821\ 999041866955142088523290767228017082232562225549519585561454743273124163\ 677752884615015405520621676619525037655045430291445732440457391207032265\ 133872885973455931705112698222524576603845551531207139553246866884116102\ 879720109627018121664051485505247708152900222204381007082807611487977290\ 022474571242269482984767634643269460034787705511869488744252307203598158\ 801208797131456945837904454446281360871712513379867643860536995579336625\ 708531291833361967074767939220330373733131582101098109185349536937600130\ 332852042054779847673724427210077599687594765214397758558907457511654537\ 752729382973953439087337899506726156384791395719423464360942977237404275\ 823845653804579353202605636029836541260322428825357163207708199893335905\ 363419141383146480419930153139873108895215765914586788744102088871898033\ 322034972161502986426562349140764603601574014173685832260878648956139033\ 253267179635276463711696661722397960059623251589849730922301109974139814\ 225892180593676181655935470791618776718384894352447190510780305070436461\ 640431074891896675080961193408764985596823401671850807319666407052300496\ 694775150925235709435622504704689578486563518471318250215823333339227897\ 223492592699276717408804984870330347634978028725490405440077499557663828\ 546462191913811463534536568835885323620058821327971596833948403525651640\ 319034482226664501620761190071107442872461556177393934253355715613384416\ 613591249106410904768861850246910900861262213935636611848690368827475409\ 341018596630918954295945067085939302612675030377067845094605356660941414\ 487828398919487928996733979554492608706680417288235446808844703348758956\ 431958722402601473999862568750135731228777986340158354155621481330988876\ 259518961366202386110907731512372731888674229414372800854375414128658914\ 674976624949902137724738543620075252318514374081771393164539644295750791\ 317747659971824278760189689678765223696810534640419080445982487720862309\ 706877546258592537621775345446450990457474861635219343443174527601382971\ 756763360166235328093526962999040692861749247190653172083972459499293155\ 045669212893614111475800685065190430930503069439247134005226897232121070\ 641516729447492602378693624905831730795588964493731641184640641341414536\ 054794386641381734077652953448678899728987056343550829337469360919912053\ 743235403316469901898105111400740961782855087766303968441704045969753245\ 275089730463166652758448858270473852442382549040016185736258218638915529\ 977172873777873178351372499128677248974254952624894064345511848170184986\ 317266488242533226450697877182416733404128767596266824158735603048610762\ 764544493047638289662857389744132666400914975523386685279812492989632591\ 981919210490884087781788271954218478368597430226995409718641627992538955\ 938092507318078946663102282572187982273520500934025297591969854319053237\ 448407593859225356947375951477041980630619319630280391822170699916033729\ 973531653640923365374541110113549730759679601215189880816019033378392511\ 679149305391442174570439628169290074285047996542542232531663732291376929\ 075396715308872198836160584852186697401344425082120086726458115029558230\ 705905060120916007940707593100788399231011914616029314824254002168338861\ 597208128272605531722929152610198606706924004692195017810474127104680118\ 886664643592013043080894876843375753760349428668801328488764292685252702\ 691820239016748445837148986723568402108618858575053387514093666975220591\ 653069134967194994506626038126002858073223206637074143384763766994401658\ 927515851581648327422064296766658626156334713909239123447183631337831653\ 219389544110164550290631319273712942804976773029044444848015644136277229\ 068498594794403592905576733245057811462640751558936280181078512374912969\ 393098795078409915757303064905187136293226147194948874289168890800893715\ 595941836032729892449744054754096959711146134786737491785463277321269561\ 471291693501628639720645028534935751714495505922588737801287022313899760\ 630929703691498145679206309529076631536838457504486010163471232190035520\ 614645321927713968298730663136776464422649323122820545409165309607432445\ 371185036831400995048019702433340678415558845526335770158300636391311551\ 263483116592239735839087688723911464624533063633170036066957789974910953\ 876998963670720009183795740347354756974112057458176925892229411788336990\ 202617225920692044898485374486776371627579635643717021992284320600662532\ 112905467686304310463715901300057773366188780843288374098013229319242735\ 568676580641803266464723439835813184276466364589752936961164345489479908\ 486072389347057297683442296883746134347610924593219843449041142390149129\ 992216539163159732771219171344289573638399245670206991827519410234449644\ 998845760979903779632046725224910775549498576770846112372190394527696843\ 011072107638057860779760706697279922200802931780890226523727337985498321\ 109716383325862360331103724176223143558319446262147378850570094208114256\ 688949600786193058355735906811105359380731190738806095865846934064838597\ 419522942259976983191287267848152501192106275679448230187856713233741594\ 592813007047838043201412012532550350748495917630917971122554516186848779\ 924359974620337291615791585599280451295599792950961283248057548657481361\ 474250466693275033524739539835886784271554349498554743062470240047474137\ 812394594326767473769656153986418940575901880606077424025180538605878128\ 657857561065112091140701103513936510362815221651301801185840505645208673\ 048745483484676647295380441424969372036953692683707351245362921544646705\ 690233134164449566848112184630835960569579274075809478919942826607529671\ -855 40418768814900796621547 10 The smallest empirical delta from, 100, to , 200, is 0.6717339885 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 34 34 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2346 + 68 1190 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------------------------- | B(n) | / 1/2 1/2 \n | 1190 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1190 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ B(n) d(n) But , B1(n) = ---------, hence n 34 1/2 n B1(n), is of the order , (34 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 1190 ln(2346 + 68 1190 ) + ln(- ----------------------------------) 1/2 1/2 34 (-34 + 1190 ) (-35 + 1190 ) ---------------------------------------------------------------- - 1 1/2 ln(2346 + 68 1190 ) + 1 - ln(34) That in floating-point is, 0.662572614 It follows that an irrationality measure for c is 1/2 2 ln(69 + 2 1190 ) ---------------------- 1/2 ln(69 + 2 1190 ) - 1 that equals, approximately 2.509268537 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 35, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 35 that happens to be equal to, 70 ln(2) + 70 ln(3) - 35 ln(5) - 35 ln(7), alias, 0.985980693834371271 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(71 + 12 35 ) ---------------------, that equals, 2.5055896141610302553 1/2 ln(71 + 12 35 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 35 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 35 ), it is readily seen that C E(n) <= ---------------------- / 1/2 \n | 35 | |-------------------| | 1/2| \-14700 + 2485 35 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2485 (2 n - 1) E(n - 1) 1225 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2450 + 2485 c and in Maple format E(n) = 2485*(2*n-1)/n*E(n-1)-1225*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2450+2485*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2485 (2 n - 1) B(n - 1) 1225 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2485 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2485*(2*n-1)/n*B(n-1)-1225*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2485 and 2485 (2 n - 1) A(n - 1) 1225 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2450 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2485*(2*n-1)/n*A(n-1)-1225*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2450 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2806510736269685221669956286291796341995275823150554439156700861506319411077\ 496729544547071947315269269301790364748372910077925979004635033293476579\ 473053484020853015896009089333231587402383796553033520607828358884168626\ 695214106146543418774136660553054364652726982362293970372647131900276409\ 743168026444329293673096284766776618388127109172730957602908097292018156\ 449189258138668876094819267835589896879738905086941633703111216432344085\ 852228498016705870849875353533180749599825330071770438222374267201457296\ / 968873 / 2846415506733170796940312793514362112540624878880485566252756\ / 773001786944832169657236910161706072336612818076607142680089753201622242\ 697307694711080968536172696029518416818111895562047041646605824425954674\ 070520910817735887436202453453135783601856127776865673340904271030231211\ 186065656245309516471983098721219400252052065299576405547436476147011463\ 267998510274640009886674525173444894507334949066910002095647370458879029\ 458436007890736912780783513088625126947562412616162528650238092385573595\ 685359238091495156800 and its differene from c is 0.19174218538024163599236836893875292598584926849554074965281017811678067552\ 208520473475679489373367195232559215003722439920058016781582466728156264\ 117923768907707222360317951284441034681427307336273305919522190261352459\ 932520974509980657967451214313856002635513905655012003259425999271165273\ 329978334282274874228618181548529115875566424739775951313214538241120718\ 045465283767704800767699719518986234583445495309465970904981888960104350\ 697977278466574233639407409005057253512064751308543797510337523281209793\ 976295959432191740647819433208276181380131152504050592476516756341868066\ 281805506728581855872151398312258031178909522045344949627699086060820695\ 397444523210961822499554429401080115980926659919143300890401350814802667\ 007979065900305138337495255199785370749096182446480121804692548409945425\ 566749462120546246636436550484906882739905880975679373113871801135025281\ 932689218253030014719329244280265512119987342489254461188043574155218419\ 171553668857220465527399237051439395345129417997921156556952696283242875\ 893096111581270093155207675870011538165661539146067235916258697694358132\ 055402886993709813799793264458326301769893356032546023925825655623636726\ 566111213178642407665725128358591151868954072567111758200118686824631018\ 097375346729860088146627377739017355566588999683353706067020231701079985\ 241458227935778215059072308199992238157116101000253840394661618965997732\ 886669694964603214908475211454635155668822852442964891800127033351145035\ 487601462104845158569747271442684296910580546939688746838772120800534858\ 400965930598983234086567140083240150020046484672984258810366021756268895\ 051251886857426395578559452156759517703086163943235559552644544361698672\ 785859550884843788003957780068821783485232942803530349073725740405699255\ 191995705812849085915808726716483992990033120736870943570629002371124452\ 072229359734563943810756265120945823116712921094295314277961331139107245\ 092646002186060237461226555042871523855325051672548240650407473994237296\ 785388953289581212599527166005564255049853938979529342383342635241988119\ 863296466650592692797183650829081327131511764017692306181112287685473209\ 017224190764498307083628562903344848333818069649936824839333223086333964\ 757580577336826493593157799818238789053330337518980016818599569653281702\ 785888018681769576016275814085533487625196929938560921738048512649354786\ 733824100944762935062238179155316330765906611083666958902345931367722833\ 740681074533801736604080834993678777127058300369314910504085484226159165\ 594521109265628662456681197979561629356706325133587543331160389928869742\ 659436952146094422523293466298388675451009676609725097178400827853482841\ 265748979008088155627410417558909487369932546871465428737063225990166414\ 592242138293203774602956730557205725398531982124670605088852752137306465\ 120301520047312581038885646016492135655812830770315594448647558789947836\ 546590916670715130106190040362906194160048340830233160917949307728903394\ 799371385373890383466044724840393878945781784336376224616423606984182591\ 927883444836697897718756643602333954077033500490385360202030206420871990\ 495994120295310019667157457587340127927079356885903816435782493491782827\ 580959812263936089019329259723530830185727622048891000866780660378406475\ 473459627825301569206847633601334335045313886228863509064841124727150661\ 584992944523010189500730225161101967690776213310349276047750007813295696\ 849107961103996568700529612769079481848318136603535663108692047545003063\ 266922414753577532678420950616935692341613531008769642676828863368110443\ 883331004220677259240456238161161902030289440489691097206367167965053155\ 628139229755031335853931961922474311004833013660344916671429633004817498\ 727378730191912925338549181121346842936888394876321932754475789589942566\ 177156539357274548257254714379836064618751046522475535742583594466397833\ 489885815266393109282344017883350579760311816647799461198125178688499591\ 194226571718587582177281818887939577025010475173084511712970483485400201\ 387843837479480118655645170668631553751311851728012764775541097619034077\ 412116510507148715105641034087413300012499542634718354119816059485000113\ 864936059253379996678381724592263939978936065762892560714247237896616714\ 249537458766126568739544126912600625722920639264015786484087177977365433\ 049757692033228193182968409602841060369085065674552358096329282667783624\ 495591313652699235201587727680045621375293394666887888011412516510817457\ 171519437971009602744518707859101865730418836499207371016683205466369405\ 958065345824811727941528731800786903873117806368924990716636158738690709\ 015641227405731078207062545229114802925204335147722592216880534427364236\ 774535737891812829835588628874588696879346749033328096041219656109691204\ 415846404627348947972716245373731385962737883566375006792269163544309824\ 865471631784383963769192728485914087688852703407063632814432847471969589\ 745864305609732897652347620993232644505293769755302741402021457080931450\ 896417094103936777516676323158122026913750677168858355027751053588454104\ 486971246185848951705396077232460885508979461602228372882622191971224586\ 624285976762554811509655324710691683076604533432976475582986896159755889\ 862574458787958950068486267284405832882619187254639784461398419392195964\ 669757004898552370412575886630132035795234678719137732798579244905362406\ 957108998934994749790936219409337147885129166789927174524861373809257032\ 681329302160083694737051102341440343546724504745699043337242382303770253\ 479252629756124906679950767002075654032047405802595360441029283984508195\ 306406969149456598720976058080110893020837240156328998490742828906632549\ 996166389935776413947353775702902438711902609063199913249996933735993971\ 856568193183616505476684920503400917491406219223970246530214586695803767\ 980717497638462519611789402584429336439391325318372545277585295804796477\ 669589769541367809653554392971855532109512011048754692479557064629893129\ 423412129825048238010313885670543971496587045482328861306592336837368935\ 752557598977886259625839221541748116313783383808051860766586480421498271\ 303917857700870363038227649874148307283403367115583482753931914001559717\ 661857105907035305462083699026363040278528977847494262857588406194804977\ 750336589404036397315471772082624406538927160194395302428911583760309442\ -860 818488418543017743 10 The smallest empirical delta from, 100, to , 200, is 0.6749609386 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 35 35 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2485 + 420 35 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |35 (3 + 2 2 ) | |-------------------| | 1/2| \-14700 + 2485 35 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |35 (3 + 2 2 ) | |-------------------| | 1/2| \-14700 + 2485 35 / B(n) d(n) But , B1(n) = ---------, hence n 35 1/2 n B1(n), is of the order , (35 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 35 ln(2485 + 420 35 ) + ln(-------------------) 1/2 -14700 + 2485 35 where delta equals, ---------------------------------------------- - 1 1/2 ln(2485 + 420 35 ) + 1 - ln(35) That in floating-point is, 0.664191732 It follows that an irrationality measure for c is 1/2 2 ln(71 + 12 35 ) --------------------- 1/2 ln(71 + 12 35 ) - 1 that equals, approximately 2.505589353 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 36, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 36 that happens to be equal to, 36 ln(37) - 72 ln(2) - 72 ln(3), alias, 0.986363070772119941 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(73 + 12 37 ) ---------------------, that equals, 2.5020635180499924959 1/2 ln(73 + 12 37 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 36 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 36 ), it is readily seen that C E(n) <= ---------------------- / 1/2 \n | 37 | |-------------------| | 1/2| \-15984 + 2628 37 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2628 (2 n - 1) E(n - 1) 1296 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2592 + 2628 c and in Maple format E(n) = 2628*(2*n-1)/n*E(n-1)-1296*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2592+2628*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2628 (2 n - 1) B(n - 1) 1296 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2628 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2628*(2*n-1)/n*B(n-1)-1296*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2628 and 2628 (2 n - 1) A(n - 1) 1296 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2592 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2628*(2*n-1)/n*A(n-1)-1296*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2592 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 1874559516187074055459511817174355830709264170221130993916412971965030363775\ 695039326058867710410831542704019000980332727877569764195587199678206841\ 555239604415310336651146078135278848543592265375925010548068150961647604\ 952682270026159388463397928743261866961105467406308220475977045866032891\ 474059323375590361032076205349883124394849943982979190489623614243438589\ 799998644010802121498205611634170040134272272860991120755745644912363176\ 433203999253352041651112702016605528160054935718092952481622894148377012\ / 75794689 / 19004761752887590442674198906786938452764895565132549285597\ / 694320166060662295314729956073915023996178982353459793526471238294684095\ 692802539198419590813331031534061477602137681781091011842097240479225834\ 777660151052837814276441092394263017932665618311213421124240152615676462\ 794129341400291274899515115900990913715281198901781399124248757081007625\ 799755701513606408377858357514932095691709018557566117318503066158915553\ 929315243370641862018253098285504592432610696415656969993115350345756561\ 3092068844138930237762000 and its differene from c is 0.28616481010861404204910881250910287296035588875131349954659116170174883100\ 076178404130699590929596754177786386882624826486921260486324664049181588\ 174164514346419800586048962831110283693739224291148276435771107468503693\ 149097108710533472987093848218893268787532347008177407453431133962971727\ 963904942511197090449682134362588897616780435166468612986608637651620386\ 899697415738140435708234195612852185345797120105803390072352570583972248\ 285695268242252247266065991868789414538180638737313138132666286447213472\ 600389384639363061533294991533341730350100038163995835426732578271758618\ 391897451880745137546855181989963428787739565577579680983001500433983955\ 864581671721498354654075237055328192119970554474581979518245515920318387\ 640848945485119353006727429844688794603044723795744573345365720682713786\ 947333180897988825721819656922171610886547897708571408002041666686168180\ 548396184073011743203962620539841205750864359645016002381023251026471962\ 692662221586193664116788465842713295016202485779821124598011989981724466\ 333605166307914174845525196256464285919438895691012220623116410843146279\ 770032078112472379369241267149625431997207841946190048931051872536860780\ 734874002280877684855159093197448188613752079656622456385232151348168014\ 552452045258849914440232414862358271738593307701114306464248562204673854\ 042054567897765280438730263361776856485567839379785151310953357961327585\ 725305585007071281028931676672630012038287084431304219968290745272348403\ 636125842847285655578550363083435658515922126876103733013567337005659061\ 365994970260609531378645590389178458324083323992669958726201671087600777\ 957497206083811269519928597370242639345602807015724374014408283550903693\ 599258369004164780422414081634019793288003338452010769735808646444045675\ 472111553622094917041640542120615371853301812212196795869319981065381558\ 882895562374310515911770980432299934770185692665714905703591072090606996\ 788994296518183423203634613125978274685543166329282824518844772469870244\ 146534277187055853102033268672036583094271263828952937085463181802928653\ 554082991304164916459074224961804673151545308148766163019829686207931488\ 729130988623331109057360746800974847091463903087706021735808757021024094\ 572376543379823922867975781015844331715943594756617647243359829202356741\ 244995710885511886891827955015532650393056687101735840100334661530957908\ 826194797662031609777168925377478330247554248732981166169094587664635595\ 652427885993631311118378285041777668044629913430518317804610956385683822\ 985372154158310062291748446405478310709245175388096947960657943449030984\ 121154749105527221240825582511871056899065400813187320988073162617893523\ 856514704798717129198233527080690964913681445011386945512474209053909862\ 796903894000149941559734500917244205074087598020319127113790401062173534\ 733181371395935392615085512526852931765570290567786357816346575777433807\ 826444804091203577475270725898673591022262157086800895421556337329072419\ 251625984293990707783216915449503205532292624927363387319571747733187177\ 573199484685266892602220755026637981334503477805840115523618863412423532\ 774659487814793443624234433333435876542345850168021750739412847714414093\ 377260681277570340741241889934395393417284526172409869366993855091393575\ 483285856048386926371295934353071804682953563168829592337168632322287289\ 933306004915617792588836803937997011844048753151571854357182393600460291\ 859512619750525945615853441249721350021719616346299976020336319969569516\ 089086839576312355262361541824783515413545096553297716026650103501638813\ 189494979598334869775243442082765662812288788297234421181936961103045392\ 904806172551010038464416582375422067517540035433641530220142884938141048\ 949653022819611368888985883376503216604778705813816181034025117964958572\ 096736367799409950275996620123552306686819836291639642101929206513490757\ 573375324260231294414268081940109855256822489053636805066998323801415295\ 628533317590359140735255268668867387419515812465533296692273916642328413\ 943097099040446682602720124091431888900667699104268163034509333600497655\ 590308913910280271382739110054024567512128286365977110486781982548688690\ 046178410461801847175664555177952377097992716007841413433535936459178929\ 364426925823414924952008584374967124181659936101220711194260852066812732\ 403129695210388518719552892468416008349747764291978898244811046109806197\ 061905572606921000873003636332350434018273680626143339797034588228925964\ 577466368164078488500479188015315311152364477422824169838273197439234856\ 751315448847144284171888119856948420729373955820840187677924344577693275\ 604980875076425983454326552376368223968162229032798480790244200669044143\ 524107224691183971120192969909460970194511730870210486844171115948375180\ 565191306674632753498996103916959149332699968822588425523123997256707600\ 119488670993597060918484966709964562643877690684706082739423775940476324\ 541778208366987869255460036382257859887481861930823148552565906640139046\ 559233331425438173256914851420280325765824459211045543068647185720085673\ 147263986949208032843091988053491030154585289636798965253583625089856565\ 302898829955075184245126218988975598819152709353127184591848589565057889\ 657009026319677166280835907811395449188175624135315774380966428449965488\ 081232648733241307094548289657028851095378477435105080322009646706857357\ 222283513312355066063365527445144872066398222435343947852639790023964897\ 491276900764966381763789203240262305041090829293594816766030165009438186\ 367189021169179580460754943587376258847841633965190302575521635969338917\ 031953870345275348861494198569657042996015149265517683398214646119606090\ 602107349965049560812619695243881442679929149935913818973064557810124699\ 729347829878008687645916998497043152942835548288759346197385800950083123\ 441378010966851748714298651122734936413416963395551637267456790437584087\ 605733107271087943914523467813733938368837284091973599844941038851173419\ 154301999258230476328136419720632309684413050644867345970062989027617909\ 440545481295980530233531611018150049770882878133269014297011873230100755\ 855090075489972196658397560689662167884018777957932385939513710396431738\ 939939368765739656478134518474960625446729518499712942845606350759978589\ 369519325951881133147890983740505287586824046840375395714797744889693753\ -865 9607699189847 10 The smallest empirical delta from, 100, to , 200, is 0.6784903197 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 36 36 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2628 + 432 37 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |37 (3 + 2 2 ) | |-------------------| | 1/2| \-15984 + 2628 37 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |37 (3 + 2 2 ) | |-------------------| | 1/2| \-15984 + 2628 37 / B(n) d(n) But , B1(n) = ---------, hence n 36 1/2 n B1(n), is of the order , (36 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 37 ln(2628 + 432 37 ) + ln(-------------------) 1/2 -15984 + 2628 37 where delta equals, ---------------------------------------------- - 1 1/2 ln(2628 + 432 37 ) + 1 - 2 ln(6) That in floating-point is, 0.665750765 It follows that an irrationality measure for c is 1/2 2 ln(73 + 12 37 ) --------------------- 1/2 ln(73 + 12 37 ) - 1 that equals, approximately 2.502063614 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 37, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 37 that happens to be equal to, 37 ln(2) + 37 ln(19) - 37 ln(37), alias, 0.98672514203996903 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(75 + 2 1406 ) ----------------------, that equals, 2.4986796400962591004 1/2 ln(75 + 2 1406 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 37 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 37 ), it is readily seen that C E(n) <= --------------------------------------- / 1/2 \n | 1406 | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2775 (2 n - 1) E(n - 1) 1369 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2738 + 2775 c and in Maple format E(n) = 2775*(2*n-1)/n*E(n-1)-1369*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2738+2775*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2775 (2 n - 1) B(n - 1) 1369 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2775 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2775*(2*n-1)/n*B(n-1)-1369*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2775 and 2775 (2 n - 1) A(n - 1) 1369 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2738 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2775*(2*n-1)/n*A(n-1)-1369*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2738 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 2687410480897877527754726491674692171456365018130849910147044168119915228934\ 655910092246015485605157135488062813756374945392039631189573565534244610\ 453629930356815672422464799958477425720111288432128605699156980509263352\ 432628979167117449706534048681289685512944132135724028020832297726798617\ 131200905699485349437387160344784711849254875575930080005898976310168220\ 871733306379980427433380199492808674866250810501398960279758556880346613\ 838861413702153968351038282207702227357235100909203596549221888126069708\ / 16141 / 27235654250604057956902301858273979893422195052671813059906670\ / 366154132290526675590770696041319136842215507460711512984438243113643918\ 619520020417251867217715552760586539473244604876429335513412693900534840\ 297590766836587010870534778232987953454373473431732330090657985515246359\ 286954694962992532509177633335679619211582294484989195287269411315789168\ 412628489732773581562816050873037055197271109973739565182300930759399277\ 002861093338375613888185040887390058769080650087453760944420860899728084\ 1553111036383134912 and its differene from c is 0.57674754958970875395977791693084429380338557457131226742929105091986496775\ 546485166891662653060648978927050879534224150435413709214774029646518923\ 472899724122531851980071427595250447444972687295629994411858076138280503\ 003051174341342093640737464519484351600471258966582405249920557465246298\ 070107634228496512777461489977779874756168330536294498566726899720105872\ 474455869034040504371597362167830706991349352653514016809342594767963436\ 413651918570426777326986258689147592194223194352423529016119797431635227\ 827870191990443745675734827160604792062468037750996637919450260743618962\ 013291317650757402250344330543991165009469818544558415323635546469220638\ 217431054211513072089805884532555015236246229372562689209803351199989569\ 841577302545980721319599069426521685763403576350953226257307002329760921\ 824038254837843280217830700146493776118162701008313768710370211859755217\ 025978302408800643566690396201864149899097157235967945948643123895147191\ 024171306118848394745492652156653787613029170369691278942709495274198353\ 269162827523809898019004573679452450829788263841269506848876284381164764\ 925481285582214916228354580467630827037906903448063332404322510903667810\ 530382172832183894112617105356518232430408282297690754110353667776427214\ 027138979523445199497816245651721238907344621980958822070541831127080317\ 147190321756209921634505110016501406590131156140358617897713471898421147\ 233375916934840833520692761792149606350766019872328456531203190521217573\ 201539736390839278670414615331884715577920196903890158233310419133200881\ 649286770431577718231911698294441771636792623830191021256042127430980627\ 305494004320453484465159668611644497338635457117619984762126187921205220\ 172526673157345929067478899706417138517119683366108291695007119646101516\ 294839358806503125458434300251175632464831378678046866030496978929376185\ 311619632488380656707115992652906514022655309483315387395306945373670227\ 545484802726686813831191956467859694316060844992712253531508919808940885\ 722492158313172122986004249795449717099271679837251801082303112470468252\ 779197883165671757724294753834868402204611467062822586500451273095237480\ 273364844543318251463280208916733142173992417422381965324862871619878738\ 583449433311071232400857497399562389367978637765866309899303900059011491\ 601390480141990976613627310711844214039480634664966312003676791790468004\ 814593658165614352308332343115542807140531558707967973413277497766159394\ 609209353494087467071015931856486212677376949523309650145465260432802591\ 995850005538716495752882713312173366015536869882305191747935343070900565\ 198260779294246295154741490082523383344229171612546804579292502631715076\ 192115176502140637073701145419030696248246827740173821207962052799246526\ 603724668681741736667766998325779073130124905420774811667216349594170227\ 375845249545566267202178430302292893950577431145681629429709136889871251\ 761551379051258498684795968143739901725712979541085200686599335709915443\ 268598658375714341943849023857142017491183011612805406683217376796739033\ 080674279244174608977542601444446420755646625335602642630790491062129780\ 977958670379121432700486327940261541465375456561672248463589952020141570\ 199414916490779432843471189273231002492710252482470444652139711386103506\ 696412875693817342526262184484108735828043502003794185892833993890585183\ 746299591597731641753817941143433102185083921987158087177359750947032787\ 814906490366449684155377068470196941137954499096258222639013931899652133\ 568292502682798019655191161146226348326734935944689449309507624572575744\ 210745668848722352714065313275997910040523105163261358608802738767685123\ 793721143428725451899081494181094084996106524179484208993290965400416840\ 730047602977290855537797303695058003482483827012827856881333830647687779\ 809789281962063103585836114293929360095764174034316608809019440936713194\ 276041180419060885825220159352457699456511803085405447917196966970422655\ 126977627376290494435489808383793565448168453850857228231502835329543286\ 337175887231906829068120354931902279027636546733727625834757105725450853\ 144798920302989873579518790803279014427025779954155909705744425394139043\ 858195670140587931574004124394377832896397404919225036963825570032795740\ 160645758469701026330253957896459384084130909105123479460436611080876152\ 465390978524473431319471030062856911544890132164062438252899863572144764\ 946971148568933384296945789999714299232409729931465170433684481295744104\ 611758191117634970039913397233686468690242200124802970926906150019561691\ 618452987194725657550848118292122528371873218040779457043757934522863336\ 346649671319896916184475981191985060842204012655255986406231112870010382\ 652732259906632903652192511434519805083888071686930708517782482878770849\ 160636295598565512413101596966116823518829972592331685226478815148962606\ 667534802267719757325372447862740080988909126871633815959143677709646519\ 425597537914721299279213081704136324164396654233739521806095557137671645\ 076498730507201165520753609184093442202245707903341660768815002834915634\ 512015197223904214515159227939927962268814609950359647724325503966137006\ 305224338272986191686608655586178450334842672126773911443886892286418985\ 909421073857607768173761640986841850930355543377090177160146118326473708\ 538972421920212075176852919084836875552781354950265890548046255566062187\ 120406174255846766076281526235296424711203979106521288841733738563552271\ 583336000108945085791793799155381302726283950193655176723249490376419613\ 066518881883474805858210536419250536866289825745342016690028361692506484\ 516505423152132011235822156839808173447687463443566605079042252226953419\ 198900550943984304355169577766166714640790865289522416986990848885796483\ 857441576809395978809551175048952120530824372174592741679349052079819052\ 255479253497150599197158743195146619510500886422457549771718658279410569\ 870015384866709425990371351863165440284552321211526130348998205178853063\ 163278216592554858315125561862244141321632708281708666537544220657987061\ 862677362248601725399865405961843916678096109757174675045535250677375925\ 352008809961111870949911799377413800312211129382746929518125896444049936\ 781331899793901876000818708672629618161923282606554764002364968564650228\ 089285900742579251985036761674362026277660098570965206779916769123308697\ -870 70742719 10 The smallest empirical delta from, 100, to , 200, is 0.6756283858 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 37 37 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2775 + 74 1406 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------------------------- | B(n) | / 1/2 1/2 \n | 1406 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1406 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ B(n) d(n) But , B1(n) = ---------, hence n 37 1/2 n B1(n), is of the order , (37 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 1406 ln(2775 + 74 1406 ) + ln(- ----------------------------------) 1/2 1/2 37 (-37 + 1406 ) (-38 + 1406 ) ---------------------------------------------------------------- - 1 1/2 ln(2775 + 74 1406 ) + 1 - ln(37) That in floating-point is, 0.667254010 It follows that an irrationality measure for c is 1/2 2 ln(75 + 2 1406 ) ---------------------- 1/2 ln(75 + 2 1406 ) - 1 that equals, approximately 2.498679641 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 38, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 38 that happens to be equal to, 38 ln(3) + 38 ln(13) - 38 ln(2) - 38 ln(19), alias, 0.98706848332390500 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(77 + 2 1482 ) ----------------------, that equals, 2.4954283808469068745 1/2 ln(77 + 2 1482 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 38 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 38 ), it is readily seen that C E(n) <= --------------------------------------- / 1/2 \n | 1482 | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 2926 (2 n - 1) E(n - 1) 1444 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -2888 + 2926 c and in Maple format E(n) = 2926*(2*n-1)/n*E(n-1)-1444*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -2888+2926*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2926 (2 n - 1) B(n - 1) 1444 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 2926 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 2926*(2*n-1)/n*B(n-1)-1444*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2926 and 2926 (2 n - 1) A(n - 1) 1444 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -2888 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 2926*(2*n-1)/n*A(n-1)-1444*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2888 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3454027991694957290835740578416551770647767829819083759071871152705432736934\ 915659209186899998115481683678489951462853310260111516988955103499048542\ 439386990468745211100626393515061682706775507746594543841890876145811503\ 530398194882252348729576014673013686381962621723225965294408086471892449\ 076078393871298477750602778272641963012849101322770176541013822136988507\ 412908348241278263291539278524444346063480915649072310088444174649503119\ 970189107419398230050314418769917679130809057455125172369160740893155914\ / 14835013209 / 34992789761290790976326061096493866022100472440140645922\ / 089821134515568452952214340195400256115113386579630132243481077758654693\ 437382726425715585535471602702288362819490882926036173575963222089504846\ 006936637759044602266856785188291443229534015899689555017536588168347981\ 470749446921265780137044091312991158862006871783839949358859638185228483\ 935869791503506103644080610212334656361000873595527443502289723403722274\ 964589002191863245288246168013008795617943924618920630785245013472500629\ 6603205510286473178243479916000 and its differene from c is 0.15450998362327596648167726886532248519918273130152591643356776462885107372\ 005619593077538108555790769469673120671859722172208787622423372785173482\ 273810023194504494410182001901385645092924000813774977983558472068693373\ 080520228319973549573265672142991497548445703215394313289620058535204977\ 534902474361042511599041959090707447266766367842844332723064509302621474\ 026110349800201665441506968264865847637982141895791673826390904445906831\ 301448697822640600716465983218527622284697564966196930758658199610044808\ 573946592672050207761959559267114119456731152661931428766523775271739434\ 379064274464902536432650443963627287776524438978781836778055479685162079\ 297885476118849970483906096071007975361690177698587935336224747228072992\ 477827697598377944643484844006853198911812600276883527775229728820883557\ 875028829993799486809354828160057195189781520483093916121390540876403510\ 295907969403450789271659241979327944898617626170250228978037276723328327\ 537227913029073901269897869395715921253593437957217743860649937475055288\ 243573892319915387548774186741285619451495484182746204206983997411085117\ 889834208942086825790915173527584431821694614171494832422432411416171459\ 680851963289616005300189077979427739620893147883080693033201490772873854\ 872395148981288020975118778766967677556432951247884179316370082120988161\ 162053326497741544404443197474508980539918232621345841424670797701878807\ 404559479467930000199426623063333398496485227962278163557482768536165346\ 635392846090821537419969137572231882266039954709135762966494076049457060\ 822866515686740468595388214725629669298866086352145897641174269113162899\ 361980130230677781114419783106175547832488510134815985944574013195653563\ 928269013699814315649404106353325523190115965993468313144301303307211816\ 725097919579062084306198379727895518346104637727480659351785914899316411\ 967254099942960352785575367080790675456334269170414937334386389200856103\ 110271292808593010706091101013631788629477641880320083489359952168972272\ 250289003249421122893909573313870187553245520517756416504640867086267241\ 779543732930810323398580093407426443134070925237028692582856047492120297\ 011185067472842779381346622688973128620637397947164224999637805263608943\ 865257477633755392887162948537116550028843043818578322073371812707279017\ 453454084345071136530724781491562694355228301669799305629915578007446682\ 631950495490832843316420491791969619249920957737290791109134386265895944\ 496353966880590474422405661382570673338270628615549802577145447282896771\ 643718382928403905712238722426380099239383138609939080762276704410696479\ 044096974510114852636332304853271271130358099885095346365957150474302217\ 198547157724977192239917577125564213174360280487888632374736498068714949\ 135033081199665016824183289865434262160208373004002072767889390217992936\ 416864101749521124796377027350518009625249760304183554226730324374197551\ 676976448167694107303751147716038547846698917325325909201049332332011188\ 253650992265662511656560055199958744548889714967744369947076481845286445\ 195342325674515637553587924855573191777692067562030728866082127898928767\ 380024827877732291617387652733525078368581287906725592329155597519726494\ 128767382620775832114099343313715140632932410967411131721337519165873816\ 670178501719357774714904118982532901645185365921035009657785043846823690\ 605330472099517415065739190372419552660184711366537366192855654062444552\ 470470668150050769050577878962203564647186418419806192673078375621400503\ 700614244244385831133545156021406792984092864519118012085923198581893350\ 802351773548548386032286222811236438262583553782930539955119058198324346\ 343901059832666990463843038474183401041282318737377550863969126577433457\ 673603308562677973938018401857329609138629333593171035612807497888944869\ 932330357894533796113760274418694732851153374284048055526705979208692717\ 415754217540480290793346542543381495273063997204831759328538891428465621\ 406012031624223465646160955446868898424069289331006867409856038234418381\ 492500465427163133102084867581505381907142670378161417927673480498384793\ 967757499372003327617359715263568464768688169017855481907060459925156157\ 532381276845534540107686478289613052413383245539694484944021853492092622\ 355520016756580629201004976923930053963070722075343901567785691648617760\ 921996218510983260487543468209297205222575177987570200394857916831330432\ 677895950269201937506793705407292285834025720385938847537290076189860524\ 116970176907111297739619206013204256570683343663703921304100138308150695\ 147559195682524855567082805651695887655706279109227094785526172035973198\ 987938267040225553150021822084102664530298486342432025525488130241464043\ 219496468767617177770752837727382192718608583757222350421841641966662417\ 865322314603526093535346284830040531065112721745579705687208104705409188\ 667939780455381890935986340125938281255494260846959355350162195116381237\ 292177083597767912172691959990990402438262446845400338498077209507420245\ 177375810032226895757442891293235663253687772148754299976885937141813592\ 426059812118809860942417710347371376683234373560507015057365964152260431\ 787482412812804892974798213114021932842137370113741531382518821558314992\ 407987627982014807904519345701916252159488695010788664487790300897887114\ 269248412714676109883817741066590343078224195980977205752051005467785879\ 588591699149320542153571181982763207768241065109869452471787117815493852\ 345982649425065113013342867213840612180876174710798926632257199771484249\ 833253397068980153758061839150677626807113002773873946813774798872176872\ 142453654892058091976721726773247641405923172879259198134853388925475917\ 998029906219643582516059436449227860053207909623242478300480618076661177\ 906728450459588097398226387764240998572431049552172713475157086419615497\ 542062403642540700498638928514086311500501160587697828906645026293246452\ 737242671606495703417473797814325986669831287855450108348518199154858177\ 195581040369188240746514664344540911322902301976252900536708844598041730\ 150584571212364638145058973245501204474189149862309806703688027916534744\ 693183862808993201436240010483351920716397515993724378231990766797099410\ 866917755362337920499829482394117648655532184805611988372242759236758048\ 659523666859047143442433913347387021111841789081110263479690035255810366\ -874 6301 10 The smallest empirical delta from, 100, to , 200, is 0.6786494165 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 38 38 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (2926 + 76 1482 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= --------------------------------------- | B(n) | / 1/2 1/2 \n | 1482 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1482 (3 + 2 2 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ B(n) d(n) But , B1(n) = ---------, hence n 38 1/2 n B1(n), is of the order , (38 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) where delta equals, 1/2 1/2 1482 ln(2926 + 76 1482 ) + ln(- ----------------------------------) 1/2 1/2 38 (-38 + 1482 ) (-39 + 1482 ) ---------------------------------------------------------------- - 1 1/2 ln(2926 + 76 1482 ) + 1 - ln(38) That in floating-point is, 0.668704709 It follows that an irrationality measure for c is 1/2 2 ln(77 + 2 1482 ) ---------------------- 1/2 ln(77 + 2 1482 ) - 1 that equals, approximately 2.495428380 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 39, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 39 that happens to be equal to, 117 ln(2) + 39 ln(5) - 39 ln(3) - 39 ln(13), alias, 0.98739451138730513 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(79 + 4 390 ) ---------------------, that equals, 2.4923010305886958953 1/2 ln(79 + 4 390 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 39 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 39 ), it is readily seen that C E(n) <= ----------------------- / 1/2 \n | 390 | |--------------------| | 1/2| \-60840 + 3081 390 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 3081 (2 n - 1) E(n - 1) 1521 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -3042 + 3081 c and in Maple format E(n) = 3081*(2*n-1)/n*E(n-1)-1521*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -3042+3081*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 3081 (2 n - 1) B(n - 1) 1521 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 3081 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 3081*(2*n-1)/n*B(n-1)-1521*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 3081 and 3081 (2 n - 1) A(n - 1) 1521 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -3042 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 3081*(2*n-1)/n*A(n-1)-1521*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3042 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 3877982283042792638877994817587864776727152057773324030906135682392568765011\ 709948442041594680512452618813212781756523002301385785997411141445482945\ 878718269333095170059492639385311032186382927141377954210635486878485477\ 624051512298467126658804536645328287627521275256492638462844953981415441\ 189173213544389314185011540547370837478823075831517853701536499346725097\ 145948070385540423541886385464609371676716856980519210311285707761832367\ 416943375608083812618915761584068552855966453082901238913064802246607824\ / 42447877410643 / 39274902162400774049894880471002187256652831471054633\ / 847191539413199890849304227159197857213821455503641420533356969709430311\ 499591952030523338723095160372310704950960003523708198912082264931086605\ 044550975818353641435376086620072347807934002413282061869490158734525720\ 547835283109588228915705409511997561740613323486081516089755339327707139\ 492044003715577161518513256013785338177082050656780470956606260421898821\ 850511437762564361942567642276768570906018399666740684067931413705771438\ 2822269138478678298745320848392728000 and its differene from c is 0.54223506367413799886748107483699349275780178809595870499309427669762898325\ 380101450116268056729969889387090031321490052530642012552820070883083756\ 513379081005926343875435683573406738815178670060236285501415910017791918\ 846931247407953840524278367485243626903323297427577661546404201911923801\ 026489842082731038192118017871919998255775405992137710622637382333485133\ 277051964080930954783703429887795751476892809838113715825886260208412180\ 768126050700649191836869017900301971983319734445302982963481189532964110\ 172202383254542454578685994140042288844665512431816490110111865142636545\ 046280706492218350328377498377327711907138106323946833257588673127645126\ 446743897315106201949954182181965516830096660724661857411641415300702360\ 874091460038503656928634495621408009181906877545069534928613091814158251\ 267973975508682851042319976688182599346453521887036345346009005686852407\ 580966322862744660708646555228403991476808772103684137555850693700067298\ 578719200104860088635349743970055430294645363039237443300695332284883443\ 840191023668847226730428784327537620543801158872487009774823095007828995\ 543194702063505130961997648607544712212211516555248406868359704429546475\ 693426704555658007772936541620223687375631664879318546850000153820311447\ 039076251853883834118452400561385464829155554680040173309285773654547703\ 388104834967131463721764585873450415031057741580794497896960483006119870\ 123327533568806954254157947026113570717767560650679464879478439759283010\ 811370885328677159381027770838343736950518485727340297750335473472720931\ 888099732481703044313832464098167881312419593308563906280198299309309070\ 210638143862678105295164427518528113521730044559607282587322034693467740\ 780553013803544879375258226405882646515804344694701564828000074680997666\ 940815887222346608105494752117071642744252563251590351867280737001965509\ 923209377607955261631105659049792424196134225782992118056202913799113586\ 456890506205494146076618554315217377120540392722647036248331029966695500\ 341943405859012669093354937801493086342177182052169036709194753376359151\ 869482114985528952109280797385327811639194151929240460174444642484952847\ 421248199484283976678095320529846533080119949587933341390440397683380897\ 961867908988982241239967649147273807563128583856108040375854257410650160\ 836416552028165726183995923245245083067240678946694397388703188077498670\ 678769295613116784505202175228875644587341587307623770062233955600480711\ 578535212292435504096191328470798161730476496515080625047661657125412934\ 968275088752108671830973106145818777420387649959976513271270889324313139\ 150327273025128356767423861720232353400832049813332984365074165479464330\ 863706579557984828462701699966615456014671589956927518969130349278949571\ 277125237204770900938785264194527582669131292553017985624668940459735257\ 057100526062908602862894096614878436200417572214775367810354254895764332\ 576339964409392334171170971734616257223773116647287632626431636417848468\ 675594441386354032209253670984319519089570244446494558231565325668878976\ 126348874056346805545537112278036306526003920064310827511475368765408358\ 967573340117212068885598162525555351176275493873836264546676343404712041\ 639316877156594093287951905721807829805330053189259711267324523801301267\ 215073360794856323590064749405359174035416208418515540408172452731572163\ 768510501135517496334899975497833729429010605202406625140312478901379461\ 728753269988274925036236240986272686694181798968830766050201634788863162\ 684155056095838687699477067944093023953867757899671917728092646362970524\ 036974693683236370277384771181641791966975863699530555928485869407142859\ 163880526944476831310858036635382539235159010323703682590357649120882767\ 912113597637193026562166439912969893815307777598035970781901156678041446\ 962967119242257952758165844459510243261135619395961089515701476131937298\ 144072167913791800130523621428935363604263237229003018759595401197511602\ 867063305582032243269953396556405751573745638495314793534943940021950226\ 943369959905923153714454591862237488807487828108601338503961666122951325\ 903227916799069525005180603928040902043052087004941327419288711578399790\ 662665828875343375497459411484842018956643678303009411982362316489716115\ 874845782336302700704880984442011077650321530880627782998305271738510020\ 286924947618477984358028224624293632189626444352523908562555938033145390\ 810818935023938943741390064556783985720183609421359520862664375512925776\ 347424199023998611915762035627357096022955412145163374743955335765190560\ 359506032321524056984834839848732033331009972643663099160248841497438221\ 691371783486764245275507202481558260754669388241263420700784647840548296\ 728725465939698430667696512596565828779025478617283892341231782247339115\ 205839226031110244829780773004898000567442116657113764060484110350285618\ 150668354281363541465820006887918079920126579562339096914820272346996479\ 209057734032566325444983551045411412675849377152639316535472279210235752\ 789756956732971162848453113061896623588287125479476898091313578647438756\ 890008531112631152657607457080337808095945376436477792184358214358690806\ 657144072087564418384165300467601511636269665758637899747578555648729907\ 747123938268457296966064212666849937928230457643952644966805905823252110\ 231961165743097701958894694471736044646943122874975991466626517987471970\ 751362919236948785783810231117698737144967088422649029620498107063886858\ 247023038094426399827353677579965516008933841498094469297599282133148023\ 040917862558209936437753091434155282951976038142958183259265160797725507\ 338637683141343768564437473901156037065781041675225519120431163630684412\ 940745949180358850315714872795558807583852128049764872178625425848380201\ 068132615903945108606445590005581450661893840214020123091042510126168432\ 122728736972231286217639276338411987730812565270432195563128989736762698\ 983412316998541581249880370478245879507881713146481629844586503553045667\ 860467289443320315308498482390303419756772448235839203071569825538892094\ 358533921629392721731323426964102804144179049856808784129835145699716531\ 715031073276364392665373416238733921384742489666621097867407008877650010\ 530639763696846329424718761918681968695391455028025963675832481491318458\ 30452496280994217530895822039212318107453791069604647729839759649682767 -879 10 The smallest empirical delta from, 100, to , 200, is 0.6827531655 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 39 39 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (3081 + 156 390 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |390 (3 + 2 2 ) | |--------------------| | 1/2| \-60840 + 3081 390 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |390 (3 + 2 2 ) | |--------------------| | 1/2| \-60840 + 3081 390 / B(n) d(n) But , B1(n) = ---------, hence n 39 1/2 n B1(n), is of the order , (39 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 390 ln(3081 + 156 390 ) + ln(--------------------) 1/2 -60840 + 3081 390 where delta equals, ------------------------------------------------ - 1 1/2 ln(3081 + 156 390 ) + 1 - ln(39) That in floating-point is, 0.670105905 It follows that an irrationality measure for c is 1/2 2 ln(79 + 4 390 ) --------------------- 1/2 ln(79 + 4 390 ) - 1 that equals, approximately 2.492301430 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 40, : Let c be the constant, | -------- dx, | x / 1 + ---- 0 40 that happens to be equal to, 40 ln(41) - 120 ln(2) - 40 ln(5), alias, 0.987704503614860046 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(81 + 4 410 ) ---------------------, that equals, 2.4892896658348156943 1/2 ln(81 + 4 410 ) - 1 Proof: Consider 1 / n n | x (1 - x) E(n) = | ----------------- dx | / x \(n + 1) | |1 + ----| / \ 40 / 0 This can be written as E(n) = B(n) c - A(n) For some sequences, A(n), B(n), Of RATIONAL NUMBERS y (1 - y) By looking at the maximum of, ---------, in 0<=y<=1, (note that, y = x, y 1 + ---- 40 ), it is readily seen that C E(n) <= ----------------------- / 1/2 \n | 410 | |--------------------| | 1/2| \-65600 + 3240 410 / It follows from the Almkvist-Zeilberger algorithm that E(n) satisfies the \ recurrence 3240 (2 n - 1) E(n - 1) 1600 (n - 1) E(n - 2) E(n) = ----------------------- - --------------------- n n Subject to the initial conditions E(0) = c, E(1) = -3200 + 3240 c and in Maple format E(n) = 3240*(2*n-1)/n*E(n-1)-1600*(n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = -3200+3240*c It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 3240 (2 n - 1) B(n - 1) 1600 (n - 1) B(n - 2) B(n) = ----------------------- - --------------------- n n subject to the intial conditions B(0) = 1, B(1) = 3240 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 3240*(2*n-1)/n*B(n-1)-1600*(n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 3240 and 3240 (2 n - 1) A(n - 1) 1600 (n - 1) A(n - 2) A(n) = ----------------------- - --------------------- n n subject to the intial conditions A(0) = 0, A(1) = -3200 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 3240*(2*n-1)/n*A(n-1)-1600*(n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3200 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) Just for fun The , 200, -th term of that sequence is 4538291632510054677713813110682858697458983759849841628592347639096439080624\ 684070739599127825323525555074457742517824068052325999610434246162582237\ 593031018927398165323223216291314629162795926143054112510686868602159862\ 849549484160497080080995183972215410214487326642214281165214879939948273\ 843323381026417902776740572316082951711569516759438785243008177383813390\ 093754690079301248669097739707217241673672649142765425168611778075485560\ 186755494966017020298165647165943343320858717058498267658343251173126723\ / 565692487787 / 4594786817211568254703002833100980634131367132218581252\ / 050740900642623757867511638584569690441519532442529299008148600807404180\ 026871246367928833528895400828428844663841758304860406416572048828290807\ 655980085090121725584459779688742613837928270812840991204583240916138940\ 672805913383289608282577900509613489653552296995304025634147591629185840\ 127389191703382443078920786277920151867266761266448075049363236146971770\ 477299742848355719604693836717604688909726222854023104535226063807782819\ 786181169424928357221546422611800 and its differene from c is 0.24593125973170029651685829738362596777870486641755205626595992722684063803\ 073333410738208140346501679298516088683184216544911091049193838389852565\ 036490979610009903589711416753458683907652403632639101359177228349569425\ 428266716030321945442528599160075266397229099771432456462997860511191643\ 390162469154809951828587042339340247392869967890476671630225904627924394\ 643450342722204323329670129653831183890429769434519476098474562406168776\ 400189596826164670444459145570662177608565680877853826363975932253681115\ 073379549989723541978629165663361404914425004810232128841232051960114014\ 534175726976304022133458803805727154429988598548936035689703945342964470\ 697445800967013633617165193205316138640904084940087432361610802525837724\ 603119172874768305346466132479413299044941962721571567103934894024391280\ 123419370712453391780123788174844415178793197456171573653953829441148373\ 499254540256943036120237559066047837879834202680267859193513114995573158\ 246501917351571872029798193846955863300591248268030514500028792464502492\ 746868746718618684554695515508897186640804931951314321740163244895489636\ 915238048345552005235708610546553892839635476758085539378422888157124842\ 282566747077459193989634809417600763308405387222689382658065074725370704\ 641432862929513015180950850109179022005933090733300539636889532425529897\ 218277917024355628110056006090721592461147063840889766661851483278746171\ 167032490150240880701863755942588663254908709705279059424179633794805983\ 459458030756005991545873656486196489832893300234772091187197784053507132\ 894416591425266026368496955148475660091973880516884722422830670578775657\ 086985151682843904832706987330642892370759403841102482693993311981435195\ 385833501863052851519279863745411868241841598616726470774895450308621935\ 520581141649093230644534611302292799252441503111527764774061806019444795\ 900796691092902981727989077858341617357207766005634934501472951689458061\ 393131947306663053817647897023980150652767495748095488661990711764141303\ 844175724670191258703247027776335193240692506965473839927761275331230127\ 467029658072542418051456871460947910104395858290325166773736730577056148\ 022712882743379755029226520875486745492447419304938469469739602962581137\ 010255002795694633232885110894094155738179945861619904454659392873005603\ 061070197768405264823807822223165176951050933200126635388734575062281646\ 021899970774938868415300838027900648298394508342235137311158781043843746\ 725729072067231078547281027470611272918680431738749379333192859265367179\ 719243316499099343358639457076246133149166347555997944786279864021184498\ 762232991493742399182597370300967488052253350654634178716172316675554177\ 992979860138733259361777346911637273059051246050423020153243576309996600\ 715929365566514402027228906456531322365199140844181769998500568539688181\ 422190391852283847624098683563681239848994740184221299149874620464908317\ 691346255783210078826978219502968409781789653616912342008998580562252295\ 976619290547854491889652106646223965777052443564914365558667686925642628\ 299414081800435595716476645034057571855049668439717204325679988694532893\ 671377871587479385043160665629995243114516755566431204081801141116018679\ 237808083618660911785400264455635933887495274435101699099184966370965892\ 787691006998285662776643320333336211524621618720921695840451158008121611\ 549596443359637839638042263074888298496479296096555094765706627061093913\ 030178734711133130560870246592974022543508746730162973878753091653149317\ 749051683437936271093434207475877415424605883232243351614916719428641013\ 857865436702104692670717062053716988594179700079006255954633600197790115\ 203817204662569445142590419783675467289272988643038038541381652804172989\ 434783944261732860430822474631085586484237964429992594294157303773335752\ 304895693781228795477629365876490738575736921833644540827924429889825853\ 528326634624413745161617469457063336213737139432527560255378292102577094\ 890304825921236339323114693042685827013846996155588710950539485999763354\ 774146014632053451370820811421750704459957284880249863918422865940890028\ 544203498126277621465889756048135909089277231407869555884167674221360016\ 706944352337000650423150156053688254434672746483698570007599834686702740\ 805789934614514652901333097844470295601472446569911781369882470388994689\ 747340359309152086550006031693468311017264181294859756985510273998127340\ 264164112881177360854349942838007125331529362210320496477115554349842955\ 086940213096558259826840675019219945483429536282124866124785590284035323\ 275612654705380069355175867088545090861327851073512027033365585069205908\ 776466317800392495994009438144772234049424997282150782924896835434248101\ 414339774910320266225272092998989380889610766170305575263327808251485038\ 440822726167857613956272504106463353111012330901384022801443560117098091\ 055372149876094111546258814869835276931323742661327944919068215026831655\ 262465409489851452408942815162821094334200899401026258819776488949604990\ 208241826434246384876050137561953790387211304186957063422283244178434410\ 891601366982381375837245922538636938909172937873050091628752866237100884\ 695336841223348673882583747661048325350302398080161354084089398906118762\ 675580550724514679214918672819929478487387581863083124180316606000459441\ 601144176295958690806676379447345882689750729719676969696466082737185519\ 851426614725554463992181786046928282488874129061870993131069781217645184\ 223672003756868680564249596745386588829216130819785588039247252403357252\ 674104514498381745862186974741790557930418414077020736861384052327310197\ 036566390182583602932007134232323372849223120239274664249038218315278271\ 431864694405788368494029440221586577443660040251475379281087522116111205\ 719750466511530592827203358872327233625414084045954086823138181060462876\ 574503114505618502841780240044663648706501253932089017156363613846197046\ 772260183731933681109314053245251857127057990181698756385986549382125711\ 324271738043021068716413779107894033201560616666790488114541049817231723\ 315732604132220464599934285199127780903189471388527007884972851749634810\ 058988572750117047571454294328395619629823375164040416176056205344592772\ 922075165423137483422702733609344126494226577075418650876005909799780210\ -883 8648954481686136422500696284211788299419573243110460646961001310546 10 The smallest empirical delta from, 100, to , 200, is 0.6831227133 Alas, the sequences A(n), and B(n) are not integers, but rational mumbers. W\ e need the following divisibility lemma that we leave to the reader Let d(n) be the least common multiple of the first n natural numbers, 1...n A(n) d(n) B(n) d(n) Lemma: , ---------, and , ---------, are always integers n n 40 40 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (3240 + 160 410 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |410 (3 + 2 2 ) | |--------------------| | 1/2| \-65600 + 3240 410 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |410 (3 + 2 2 ) | |--------------------| | 1/2| \-65600 + 3240 410 / B(n) d(n) But , B1(n) = ---------, hence n 40 1/2 n B1(n), is of the order , (40 exp(1) (3 + 2 2 )) Hence | A1(n) | | c - ----- | | B1(n) | C is <= , ---------------- (1 + delta) B1(n) 1/2 1/2 410 ln(3240 + 160 410 ) + ln(--------------------) 1/2 -65600 + 3240 410 where delta equals, ------------------------------------------------ - 1 1/2 ln(3240 + 160 410 ) + 1 - ln(40) That in floating-point is, 0.671461030 It follows that an irrationality measure for c is 1/2 2 ln(81 + 4 410 ) --------------------- 1/2 ln(81 + 4 410 ) - 1 that equals, approximately 2.489289706 ------------------------------------------ ----------------------------------------------- This ends this book that took, 965.152, seconds to generate.