1 / | 1 On the Irrationality of, | -------- dx, for a from 1 to , 40 | 2 | x / 1 + ---- 0 a 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 | 2 | x / 1 + ---- 0 a for a from 1 to, 40 I will also state the cases where we were unable to do it. 1, 1/2 1 / | 1 We are unable, with this method to prove the irrationality of, | ------ dx | 2 / x + 1 0 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 2 | x / 1 + ---- 0 2 ----------------------------------------------- 1 / | 1 Proposition Number, 1, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 3 1/2 3 Pi that happens to be equal to, -------, alias, 0.90689968211710892532 6 1/2 2 ln(7 + 4 3 ) Then c is irrational, and has irrationality measure at most, ------------------, 1/2 ln(7 + 4 3 ) - 2 that equals, 8.3099863401554735243 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 3 / 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) 2 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 2 3 (4 n - 3) (28 n - 42 n + 9) E(n - 1) E(n) = --------------------------------------- (4 n - 5) (2 n - 1) n 9 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - -------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 15 c - 27/2 and in Maple format E(n) = 3*(4*n-3)*(28*n^2-42*n+9)/(4*n-5)/(2*n-1)/n*E(n-1)-9*(4*n-1)*(2*n-3)*(n-\ 1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 15*c-27/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 3 (4 n - 3) (28 n - 42 n + 9) B(n - 1) B(n) = --------------------------------------- (4 n - 5) (2 n - 1) n 9 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - -------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 15 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 3*(4*n-3)*(28*n^2-42*n+9)/(4*n-5)/(2*n-1)/n*B(n-1)-9*(4*n-1)*(2*n-3)*(n-\ 1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 15 and 2 3 (4 n - 3) (28 n - 42 n + 9) A(n - 1) A(n) = --------------------------------------- (4 n - 5) (2 n - 1) n 9 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - -------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -27/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 3*(4*n-3)*(28*n^2-42*n+9)/(4*n-5)/(2*n-1)/n*A(n-1)-9*(4*n-1)*(2*n-3)*(n-\ 1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -27/2 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 1229218810335214986803956493463746059734355336697132015970626806228906572263\ 930935192549265650514056628380233919350244219853988777062639493299921704\ 818349943730532919841219587379589296830577078702284927907586546285286487\ 417806315093618597701700077363390912842282172914102886985759132672657903\ 544865680729478371433607146363876696132676448030709377213579693450376030\ / 372362502216288591287260827503787 / 1355407697867606759142459542516260\ / 436128460318182453941861091496948986866064492905223817878252953320228965\ 187688082939617029129819122371130666235906435549711153884490821841664714\ 604427669089911356500550752478229012432782861105847898670301201420618151\ 957810882602895248291355767495355911182414703397670742560226472565995406\ 116900798482021110339431283572061827712887852171736608944515825493532896\ 000 and its differene from c is 0.40317180293139737793143326926098022601636230598558274084807353584569693130\ 075812740835894097904011912745973314226823020123734877800746543509315548\ 482621079524924549098860146119175959399616234648171217877973648006471273\ 357103899777540024893534473136470636172117268370079866455862380865038983\ 748301337818290163127173753582182783259767852141155264958476613194090011\ 117396652464499605271037036242521699482259643396536996081055518042856865\ 734606264191623756161613252856292981689868206414471157178439451034895168\ 470544456956777068462364344076555374117329410162072031116927560622295217\ 283695334867382115993882236498826950843495444570200032194315871861450312\ 434894852564164361644991347259878825824141381877953994904101527694272350\ 659777726152895646010119219971940680568629196332629560953157907673354102\ 856336606794996510754576482223518401120836837182302952937802298097663494\ 150917435109963877953967097945677939490914257960844010391010858540154993\ 597574762591382985737884148704010316401443022165209630558227994313659359\ 610968998548406027996909525622628692314223075879064001947766891963625400\ 987601903870953632876811798289680119938429542676401660398491630729641808\ 555375823940596323076961861481815498951350930282697773792286104829537614\ 668339921334933898635711054084240920819266844750673873141415901108398082\ 582568701084676482453383667444462314688695015242389981833289198642232908\ 471077086935979809578674681560045502032483797980071721239966843964737024\ 182041663254506766844564736036907040875918201350680516356961722070847891\ 800039302785136094421250853511237512808259518948280464489400695091012594\ 384779797291468233844251213577273813605123719669908574307517766828164192\ 179021364819144412050765620076269967401871730349729834562556987139079591\ 937817479798863316430232440608377522633420104703366006845866642368964599\ 054019887986149932437824569176229759441231126581324271606839626234805724\ 052916425568767081366375158494882640231698367405375152333937458054847151\ 532264820901213702877061201543972841704218322621482789982588953713506173\ 617782519288815280800739715797215762338557381930757527041250702187919315\ 057737113244823049702989829812107705734324502981031622412202521653634578\ 212982104184299124072531111546505080844199651765568031479833606827524970\ 291727431279494911063834423654108632067195224919436380462333670408405718\ 152755417913298855641783652451819074195016464441976670681648325368583715\ 834154912075971631515575337001498678423388417055811363722072374544891734\ 151000520779248309378620873246615924567333872662837051694862856847489571\ 799004356740296548595702741767624971673477445584632288302987729200003966\ 553599423981315824460855047529140070525639748496187968329678105601286720\ 887383821861554198556168644322482728954431809620492219544366330737302893\ 380420901815688184153200084717454622729960115994846465413428464843338867\ 472914553255780798641855330038362111944111107596943798596427252418989606\ 032078015961893136953926242122396199113447927079636636443036317847453950\ 848213480766749993529103600885636740255572900371983653230760473515429658\ 172905751662085025692756821359981032746909176811688124429808205165667356\ 237317880945137839902844651688571493095174221431343097182658875017908397\ 652058147215227303148896218388063656078628434783781019474920257783318933\ 067160627373147890192188261551833175699725491151817552268646234170479466\ 036157627002184937602344516761223486348690842664817381499718573539889910\ 234752445585472530388868038430966373161249938558524760183332923034248994\ 192238387122087970051800474017024194415769763174363223766954553567135985\ 332715101094783060144496212895560634230192279797165303960509557811846364\ 143134272956089122486833887471487198390037591693912101556082331990073377\ 366954649779318026242323075006049029801001469283415811476488350687058047\ 543794045187960539487495326600643696162343991400054811301577727959649852\ 222702766046290636514711519222427019278946652052921430388577587593818703\ 130012907314518921619730712932553148416361405086361718865890695234488475\ 618277371112306848308631078181040964299151775591355425149476570437146032\ 039564983421682272520250115949469915650316892321135838051911309782641399\ 592341150766873456370459601794813096608913941074955594730214672779619390\ 253267583189792541679731207048638950273264478062487512702258628831470409\ 814080540497169845476384795121427997262539885343453144348191207333166542\ 243128691353081189783936122891938576163343958169007805409204710361462148\ 850642496685737571412204003609767301344778938690731671589258988011646432\ 999764325812873003631170253341276439073358003407039482124678609736656728\ 582111276499267045304218202366925071536964469053410576047548680276299965\ 597975583189789662851513597985274147647774739092065997650285198807676433\ 329551107789709927981153380850336058654599302120306771360847459451336023\ 472514128306753925939801016389999940227578229950103042871023698116435617\ 208479437323240424781963546638293268289446531000582229875084669629852184\ 330139915912417186002139085965797980286769644713208297993805661201590312\ 772634942421228363991406434595990577249926340653016230278219564933341057\ 177550636697601268127749311096645611106169788561297308386004299445504039\ 256124428105844890060316493045352703954004790979058661119896584519923891\ 739485490530400459138006802477454405603720250997084426512064784498810219\ 772171261256869544963428823257153628602831900707275504499913915145253976\ 065654709367819456388753651652963563949969126302613380958310352780031965\ 400926486864766053603554381789525816915406343271261970244199957257167600\ 768517458158470441498431449960926424280082283013633160744871536331919064\ 270263639982008163045666284327766639838106779509284175958547293602927882\ 470031536142497256768767028803526959047963673261707845054293845824311722\ 345907335479144593859312725797013863484536371366450793290860504803333509\ 831448448656430987802699799285530989833978602205884121039146398034783204\ 452650138701307936602966714984514904002193809783592289933842011330705579\ 580601559749451869829802263206820577630517087193830163928368814567177952\ 310611375784666520424369227305719882289470529956327213271655750938358957\ 937333261363775991239783369085811667802583703625128867285130473702626223\ 136777194070885121366465075151878548642444889376069391836927226838876075\ 302158985477744169849358845181744064454592991877298887518622090352780717\ 345641546257142978429797787669305848233037135012631495248469931840651979\ 460080615525240084979946261081958332581331890605401101897387082489161894\ 665015825290359583511606648568440325375099711385405862871643863605788328\ -457 4591472842632780186516898405629116103206365246048022239979583 10 The smallest empirical delta from, 100, to , 200, is 0.1425612770 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(2 n) B(n) d(2 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 (10 + 4 6 )| |------------------| | 1/2 | \ -36 + 21 3 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |3 (10 + 4 6 )| |------------------| | 1/2 | \ -36 + 21 3 / B(n) d(2 n) But , B1(n) = -----------, hence n 3 1/2 n B1(n), is of the order , (3 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(3) That in floating-point is, 0.136799155 It follows that an irrationality measure for c is 1/2 2 ln(7 + 4 3 ) ------------------ 1/2 ln(7 + 4 3 ) - 2 that equals, approximately 8.309986674 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 2 | x / 1 + ---- 0 4 ----------------------------------------------- 1 / | 1 Proposition Number, 2, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 5 1/2 1/2 5 that happens to be equal to, 5 arctan(----), alias, 5 0.94034336056763435182 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(11 + 2 30 ) ----------------------------, that equals, 15.607845206093606160 1/2 ln(11 + 2 30 ) - 2 - ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 5 / 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) 2 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 2 10 (4 n - 3) (22 n - 33 n + 7) E(n - 1) E(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 25 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 40 c - 75/2 and in Maple format E(n) = 10*(4*n-3)*(22*n^2-33*n+7)/(4*n-5)/(2*n-1)/n*E(n-1)-25*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 40*c-75/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 10 (4 n - 3) (22 n - 33 n + 7) B(n - 1) B(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 25 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 40 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 10*(4*n-3)*(22*n^2-33*n+7)/(4*n-5)/(2*n-1)/n*B(n-1)-25*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 40 and 2 10 (4 n - 3) (22 n - 33 n + 7) A(n - 1) A(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 25 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -75/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 10*(4*n-3)*(22*n^2-33*n+7)/(4*n-5)/(2*n-1)/n*A(n-1)-25*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -75/2 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 1958298536499510174759436470118378849067531241742651967635625678583222606898\ 942608786475235627194473143208257630526345133447171158250042928802950948\ 446486704139256787222880554315077905777509569525898904690498531283338509\ 202698612742455129235389506267292144417867228261523629460151825748315905\ 155162004556759086237499717477130502007252503230821364401992442772810418\ 437407640040950286411497549215104011663348614537267549089355920756380973\ / 89039661832751655671707802726786118740437500392421871629207 / 20825356\ / 126483324369284351598248191172133463621296924914562171949204003432303970\ 996716070593068578252542571529514475964614357570495734648753784139944708\ 001246566741688903968351145375001734151133144564626386110176423322131653\ 572861446756841923617289615629913925387703066889940726992510352515740665\ 619939233170024917846987980787830391650395145492194596361933518626560854\ 820059867715952543398116913148792727672488725948645669552567824256848644\ 5574821962168407770253339017269973700083509059521174085 and its differene from c is 0.36863140560577660413857613260299898945983220210899024165196931384785928410\ 470362844964613203817492597913210210949135171090154186833781968040012254\ 976731327179834779653595540510828951835848555974742575943854253938149887\ 528171050727746054598057789656086603961215100069589644172559751392963595\ 895137254320884631641766844684604027539680589457674954427993750012429993\ 176147721708400409562107991583071944405493859644132914909780965875439192\ 650618511729369488944597193397998342219953136933624624037247686804918634\ 911173831453177916174729283103115343681619260460989783964217587514947732\ 630771945269060095322415207929309359009014106081550903353650418773960238\ 412898887264045716630630207937905361390689732036147045451222722344959044\ 618949521181541708854845896057922628890867903287307608285710772745464337\ 625260733748353451209709321214307367578990015987222640957361149908076248\ 323968798480801503183049937048971262545295493183965169007930951288596297\ 348696750203265528028741161494996709771490288115528146503329202897513594\ 096695272953320040086493087510631768559614999119124133322540220000003995\ 838631311019304139007875317162236423631972183524165694037638258380820735\ 393022946777694698760370260116633992269857439632017927324730991494847746\ 992407194282896688952793801750759261898221423059437796037292684432490361\ 418110837831635104152731453895614098864877969073533791319186978469832897\ 421142825663803417225211456899253958063523618714935276711821633614887257\ 549067226936484624510199707313391105207552843799233506477213997981024671\ 993049514907919380268509104805035010339878872009529670422010985453077931\ 058891308660125756909819072053800306889207753956870799460717096940895839\ 091294051057703345369380648664999369290953475044851728721259268405080985\ 175126258193171430769147320556785550840477800058331989484618177686334837\ 733852598264132783095675466089956704028316530614067599987622531201040233\ 199620501611670923738505873502718036575452430580188227262874794916835222\ 442683280355268389508288353252803810290097274243065910631788962685752386\ 994187038620942862091083650979132565654500929523782602214550058268930447\ 864790640562973935359485406565884680016034734497147290066863394257491615\ 679223305799124678367536501350598207508357672892014619088086515236971376\ 363996270639702766155984683209884971599146657011650128603566866879053661\ 317407377709984987416631479700836577648895002933422839100831034025953174\ 875512416960311039470459346192189002191251561158434167553015880013658512\ 488402776462157639218227127106966751790659525758727394982809823856400853\ 934198209143909101789756691648125027129264922804994544470654544594832851\ 673889248793335776308379643334642869347055853981431991178691073300577526\ 919545666533812575275717801646879667075417116345224460692353750912657631\ 681970810374419121159511778306251952876449152284040677551707436585472561\ 176506744002136888052466115565556165123797523027108460101079995189277651\ 542359504037747697610720050914305293169347391437206789543771648380262680\ 931341512683582411625583474621180688795609272420529632363282362274207547\ 160293606319779813247421246066262132567929357482263215269702587391251870\ 309403714373121829779455791695230304954821742457406936366186242788877404\ 012722451846959267007903936842505236804383183625459683847472376087488702\ 225767965112261845108655166127351568471372707466942184012300012832266903\ 407835340209471838316992990175444877684900654873700237317927294241550833\ 250412356454372997133626402857675227029729271099424240285947470382400368\ 806749038922123130497414241734998299099741712528825346863825897356963565\ 648846799810951509624328285170964962165841207330854616430726651867360767\ 588248651014922650369547858770322130189400539712025803783896144793627167\ 204227812661924149138669113699279992491619613678439677367493237073312751\ 179751369646966434465202283000119275576342077592267502422669769444014920\ 074777840232045094920959898366397839837939921836741249653312488786554989\ 172303894378799720347515756945068994855337743747971829166047162562666359\ 588209229028533477541470297391710453773617050702396545141048675785817752\ 222096929898299533030278916095985403855878613868325407126736311487849036\ 687501036976809354032385418964720676278177114368353072622528355687765596\ 620552722474819115611651093341540683134345851789485549502244907318780778\ 367063276437099502141291229538398618452587257035513440042604513711251899\ 566636531473728392444569864696881878239509220623762195994575903269052531\ 199783040678882199498156879158500845544402100415604419607470426927217483\ 559516459483559785488526874679399378709937036537246060635929636019589161\ 235253653978336011551549723032738724673029300600960282889018467520750983\ 357789065049283397996388694805922944456069888496744411439153130437973347\ 025540358538267905704819879066035194355128652158788454521417197211195173\ 107372504622823966113037909038628927341074345619238454808005029459110214\ 347206414837129416307425422584860257169066485566805968485680670948219238\ 265686882526501226772557311917611411249633691844190002491488764215898890\ 957553380491375077018214639414782303211008957596716793413846091570391686\ 263296488447483099973583355648179293594614478624183155482597363075669012\ 959660156994534816981852817436793466051165843984590565087638202730771470\ 467942426971394189930687553532183820007505100256195674576899019411591684\ 102019015186218623840901309827458354421366360395257430553002502619015861\ 425175201676950163009754351294222324949302220015083973569761819327492659\ 320676941859937379407473618508143588912691758835579185528671961434071930\ 655159231156606175061665924608799120183778054900063206273832631962444918\ 365298772023176740443332429144239388719831843639586018153656015510608686\ 768475028276158224028390424697939739136817964128828201415289385297664395\ 809505977194702659790507179458407355248879976575359556390586957912975112\ 543644553215398879902902990651426920143320731770242654226463559715149901\ 108997586982969559524507868144466405299911299658752283805955328257710746\ 272036599079410118963052496071029066065154456286026774665302548913223448\ 196978748598559758362438105332749457033320329014371672547136007548248395\ 148655170477494040780084996430389890995562914320260592121782236307137855\ 055362214259881038566707446746895216580067354340092671431719062636616912\ 253552507582686088841256686198584127154781476596775444667510321143786152\ 923370259120269507160772635586713104535045882921327394968447526347609772\ 717579831218037941748395571287168295401678750928071010167792216138722863\ -536 696906199966318577657553989497420045284422072870724678 10 The smallest empirical delta from, 100, to , 200, is 0.07960356985 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- ---------------------------| | 1/2 1/2 | \ 5 (-5 + 30 ) (-6 + 30 )/ Hence | A1(n) | C | c - ----- | <= -------------------------------- | B1(n) | / 1/2 1/2 \n | 30 (10 + 4 6 ) | |- ---------------------------| | 1/2 1/2 | \ 5 (-5 + 30 ) (-6 + 30 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 5 1/2 n B1(n), is of the order , (5/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(5/2) That in floating-point is, 0.068456366 It follows that an irrationality measure for c is 1/2 2 ln(11 + 2 30 ) ---------------------------- 1/2 ln(11 + 2 30 ) - 2 - ln(2) that equals, approximately 15.60784524 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 2 | x / 1 + ---- 0 6 ----------------------------------------------- 1 / | 1 Proposition Number, 3, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 7 1/2 1/2 7 that happens to be equal to, 7 arctan(----), alias, 7 0.95608754185181253425 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(15 + 4 14 ) --------------------, that equals, 4.8569705938289178017 1/2 ln(15 + 4 14 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 7 / 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) 2 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 2 7 (4 n - 3) (60 n - 90 n + 19) E(n - 1) E(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 49 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 77 c - 147/2 and in Maple format E(n) = 7*(4*n-3)*(60*n^2-90*n+19)/(4*n-5)/(2*n-1)/n*E(n-1)-49*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 77*c-147/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 7 (4 n - 3) (60 n - 90 n + 19) B(n - 1) B(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 49 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 77 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 7*(4*n-3)*(60*n^2-90*n+19)/(4*n-5)/(2*n-1)/n*B(n-1)-49*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 77 and 2 7 (4 n - 3) (60 n - 90 n + 19) A(n - 1) A(n) = ---------------------------------------- (4 n - 5) (2 n - 1) n 49 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -147/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 7*(4*n-3)*(60*n^2-90*n+19)/(4*n-5)/(2*n-1)/n*A(n-1)-49*(4*n-1)*(2*n-3)*( n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -147/2 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 5167908020147421454682992834820918547706416660612546737151135529012122585914\ 111543128729444580603926938784130378719060905562169556452004556878268351\ 729478705216061568910440065280639663189597720624674895597094516346117308\ 585951580315632406758170132968456298174541402945107492678460593751656982\ 266153215698583105412506700491523638904365544756064813804163634257908603\ 584156973416580657760591656081950577772709749776838672952238139877700465\ / 975185327759245038552323 / 5405266561823283550502642852409900536402684\ / 444320713508664449443163316900411777418482117818889783409848541790077623\ 937225959403763309271115726329152921119367078582790065596861711339322044\ 356846350231123688665564000856321451993877336775453339910773624325906856\ 559091319969265855419316142460909470999274466915946686650183045691687499\ 246810775133968624751055788111974622104301925266392151651235482202073295\ 674732767913087749914813155765721599354474363560582624000 and its differene from c is 0.33571189935580841502522348947420965383211791934955236793474608321244726885\ 273995866987678628789141123426011055745687717282054200377991266482428148\ 758596242801247172348027459244833555717406126361877628579168404876885167\ 843526998980292508150568275064029159954647821642349024216012806054502271\ 105349444902873932994130272482277901068936010996284924051390749872578645\ 076504197417099768649415222067885486636074501213163481716220495446962777\ 347567425472951072188163665334891890567447246253060730685419001669429837\ 473772512587795088859816909953117156500138791860565959153201950758394651\ 090525387865240627773617652050166845403928075961414283674023909946441752\ 354596143252042848990361692265664758748772328188270493905787098565663847\ 169919464886970977398367036458208745252743550463059560052683972511860098\ 588534272326404740590518435697837037707125344292198727352671347166507620\ 423115094943804045428471669665078490161001284039178535249867228274151496\ 639691860752806669521821806785414013892348528200052580289550272539155964\ 590371837050110136744511448840719290840305644297025693163550740700920517\ 435358952363291776560931728145318558973217187853364475431566370512939896\ 064270602431319886958931110031537143382323185524343845699975727604211629\ 171854968105071990925603407345408042131906547750317262398749273338484416\ 244971144177868013338173125027628753089279484784392785498763187320997652\ 950899106353710135745397986618830819904092476912389285063659216457282806\ 125257299406847743325331785399036223221470397880439298179051922233476576\ 440331012234782986419349737509873833114279017356065269014654429347221272\ 337292552640775613619623500059143639352452669882829470317698742118559993\ 758983107783204256171362491282374500711698150566992054010131740432427395\ 719614699731492680716625565888715487722862120911497583251891695666717584\ 949843581331761453537562028406754536276790831061907783934944080490917678\ 077628122491688990919657927685195378549229141021821964063587794981760677\ 642625963832682754306567366608178385825066020108688082631833821032713244\ 320518813355527139090190048024648041490799527142409418248358776810843859\ 539654414852228789208292713897859723939860800106771486032744113497209328\ 732071338585941916619911595833445920318810968816052787241097232691504709\ 952283887591752745157880763194724056146169760869207418636673436381685452\ 585618822868471649873340633683222198651823221500018542426852972868538247\ 338570182757954344348935348540198883464829580873372473281562170774426176\ 740483584211486871976797260133690511530705623424404165379571001480579298\ 987719775666734550926319557834421831581134038070281657379957117418871345\ 129868610244239816612363965835920169519688486883392362686253119180763245\ 195965143195604985853958001974643934147479364995077017382821562560440315\ 722946799821078419530425421236972882552576220304393141120682587954299348\ 364693313940929103738361773795857214740374735108531750809536210058807887\ 942110019090094448781175880536344273430762974031824074581054398773794444\ 175632850218439787518146913928256577007791130776944687823608536857930823\ 733193705294161545704675283646820136560603549131032414480175359809606609\ 358040498828269710463841643176426221339288927793890497496839388794733980\ 778009085250649337937726210761258516169500751479695456366418749520873609\ 970950228143466856578365393028849755369631303476302478636509157573722485\ 361626834164307497167042781648598204582736185004247152640045648587415127\ 461383978883195419411747557610502726438612021123319124813198023220379566\ 929368544302230520666455737080279100685094804881343163997208608055485461\ 663543521213377199677167306703934425610914609951253393688292394327855659\ 994743142585115335276318907220900709597955821373027586726956233083621446\ 075342398866709975491369393915331868392000617292401766473148384583588097\ 219707429554995454941218734628480432361270461487320062180655973122122840\ 753450636603012921618234358653392574894019523011460341599611582024509007\ 448105499291769084039958996308190017693033464683280992369545553661059509\ 291312787254446150082407599484055740264070948123492231254979445566843298\ 982393960871706852788311669418077878230022995433817452124591443666754834\ 095990347073608670177757124265286230506547990939248686645426003449637436\ 861743450696140143998308510016284630329564201870044641056092149625851375\ 040870453242406376663385106351383783309397210002152596946773580623158804\ 542897952613011068029821802433229481420915758512416704778780684090708218\ 398932782979483811096387798049775931518593201794016351888883066923250774\ 853459435748488718868652661704620218415220991203607142037283350505124732\ 445281493592433049320625678286112052222783846696263535993899682152042519\ 954172355985250263254447889362601763126194995370008837350130637350176425\ 022346420337809317579624621672515861624882889322770039894201284371601913\ 867471412974093926670808324585401366332866498255389499432190506038884769\ 280559285411570817001608734385126687621717560421830161197884454076693579\ 412363879696374131926238830944644289352452496329484888052073014054399899\ 681823099746065687099436976569901966038025498675943204371194473626434193\ 628964639506897085282155579675155361612800905455017259332553876346774536\ 002025614723563606580681387654224629651851640837065360277892128346139051\ 623888762488832068858649327830980412443430947350787725152740489564035044\ 906423387471602700999373708227344428564419413495971754345570678587293312\ 009144062103175487139248350413948383036130993716460009576741543524800505\ 919999282789561890094340793903305727190071343778059281600287825104364174\ 672496119687741184539391663989224334500422856047955785041349142253781106\ 778554319964511975029853858525772427447295872188943648342968538147346079\ 518723630802671410718038759473312887478646004380604593888753092172845865\ 243585973539246884017316726564167948504031398110398750171222496407245338\ 042483647779224593586551702558523859704732897162199506694833809228659558\ 799805495721748246546165592112949972233187854671965586222822084532993301\ 686524403540014101514101620057300446063071615431410089201447583157015458\ 034893654192590107084569301576521676975492741284079945086377988432575686\ 382670183968102095000339853424120807651972722560365180628184191949911078\ 555797340930685327080591316557937590971762007695781597745577826355412390\ 118484440167593661334254495809426778586439889579427706052422033874275282\ 432593429951640124488164607284060863897101245799430209056862663120728723\ 462632038589269022617691617080555783014987832474883802945070507980850529 -590 10 The smallest empirical delta from, 100, to , 200, is 0.2679166874 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -392 + 105 14 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |14 (10 + 4 6 )| |-------------------| | 1/2 | \ -392 + 105 14 / B(n) d(2 n) But , B1(n) = -----------, hence n 7 1/2 n B1(n), is of the order , (7 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(7) That in floating-point is, 0.259270841 It follows that an irrationality measure for c is 1/2 2 ln(15 + 4 14 ) -------------------- 1/2 ln(15 + 4 14 ) - 2 that equals, approximately 4.856970557 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 4, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 8 1/2 1/2 2 that happens to be equal to, 2 2 arctan(----), alias, 4 0.96120393268995345714 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(17 + 12 2 ) - ------------------------------, that equals, 50.653659172177510441 1/2 2 ln(2) - ln(17 + 12 2 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 8 / 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) 2 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 2 4 (4 n - 3) (136 n - 204 n + 43) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 64 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 100 c - 96 and in Maple format E(n) = 4*(4*n-3)*(136*n^2-204*n+43)/(4*n-5)/(2*n-1)/n*E(n-1)-64*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 100*c-96 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 4 (4 n - 3) (136 n - 204 n + 43) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 64 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 100 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 4*(4*n-3)*(136*n^2-204*n+43)/(4*n-5)/(2*n-1)/n*B(n-1)-64*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 100 and 2 4 (4 n - 3) (136 n - 204 n + 43) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 64 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -96 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 4*(4*n-3)*(136*n^2-204*n+43)/(4*n-5)/(2*n-1)/n*A(n-1)-64*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -96 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 1117001117781471581041800359511342381451982884980505527921329457395245284657\ 006509560662625108432065441154732786651110855708570404066678832481903213\ 733324191208372790927318098222105865225172933798762542121046520477226335\ 108029866015689023044011405609774019897983233090946277036659959891105417\ 899167490663578889083625492064081458841197998568492309868082221225209938\ 736527681937220247020638433137820453808635695170033838826951551447951371\ 305701350298917981393261874298789165444433363176825865568688680120148641\ 145141345150116387113329799444564278525032470814021906625676963037151305\ / 7585304045224 / 116208546364923390463844308298867882187881089658555276\ / 916052325762942051379801803314861508910704391713585068066603621263002440\ 346891238539350221767751487459866307843779927142797617684313089560758111\ 570873544680214961094330348640235621557699798587034871632895168995114777\ 709965729447405431042881063812637764834219237387850417475755335489434285\ 775159118835701659848061970136705181296053971689186048285113624985396370\ 882332160135457729567158571248713237700060629035857330271979763523866834\ 907426448870004874180474547859830396442700639084085102402083561907559446\ 90023623982358682279145291620453875 and its differene from c is 0.55181228392235620085412506985436131324283582131409186438515434039903519872\ 694500258783095118876632203299343824855100647295351090036628743801762598\ 365588412967509733964924172319996756243524952972922152046937214533444974\ 290273722114204596364139570916594513973207987786162856406743037828295850\ 117113237469356585429379250752102886845926698707549344463749032970134839\ 009792499770990799901322807525197137110344934284775571750767297720370423\ 950539570737186218836169702770961565405588687923103814962380525879116840\ 525894531965754527389313707665074572344856557984103344926454522464729233\ 857679397732236550911914020839892316076356186153265192003117670984515740\ 100612761518355720748757969941489998778379875137110963383567228291325872\ 431424629044793385406678458071756555863150814915384043860504600528157272\ 007131289173297137355813803005013574680203691513470571301749504130651454\ 588203981779640974103654653530861851798406017655129564584668491468779123\ 371824960023235126679279577068413335264870795356687166121987038899840282\ 498499132511130447121229587951905606070903173428401936199439388088795724\ 259331734948527479252112339459697144726077375269867226402384498648494124\ 003773441374006589769738607865885642169769418854871554379432260095255940\ 964985236839350869454623144825300040086068798570661778541229997080203366\ 214681968896748446543548755276311776375528050048962743629851680102388984\ 545903358521845324032661340319136435651504651225731488280441721599851542\ 928642436176841956271199298919146939339338353087875061222835035772657738\ 497452160494886325296235144125651074493154282993645097704612131957825839\ 730192375797200115994088952085066908187283199143404812906044800435425403\ 922519184268035867099158100863735510000089360327275416331376097437853595\ 908901403117186544783322728652516339180067809962344577105603116016158577\ 251154432575120626506811851991318767241879816255481017863111474189828648\ 896583524828933356903647014263757142494952085179819220905283763198124591\ 956189847025370458352913092097806951869211517866605772960445511214882638\ 037105691202374800819145657950325014492261066235453318614054848628142473\ 012712019408199306382882280281957456961452897866530733914601217917992407\ 215731082939961581205145398671156907010437635901270054306405462695148747\ 688441685709962114078985179490111922752385421538599916012625205153523962\ 170646483330074158014748777675080090234492307697318051940502695367097028\ 087637984977629677018594417991529293622629961426401842204210190724340435\ 304432970210293234719535466125372094883987154185091208716558857169287955\ 296372474765513112683780776845649879443899816069290812130954564513162821\ 881181067628890332233123450190451586813381138922105944374072825946249069\ 143975366196617575849203679880525090113903003954626621934279319209675754\ 828779513216385158342681434223121599834460908555806979687645919001806248\ 746431050017584832617782274557978617634184926516947625741112008545257956\ 997541238155391506188103987031551183979675715715046596032934082974762205\ 709876080023140423898395998015909704035872175254343136401586151422710789\ 331912904856407408500049985391190582464460282652175431067842509530830209\ 119486329666793149899403282867973660206256721008899891672024169880530031\ 118314986442330361423620953423859677598148734777058972480373239963913465\ 357265322437746541566628405135458235826674580174457365749580130274377740\ 318268381996388811897077312365078585820840026394350315511908724237640519\ 814111224427176272379547000596518781975965113989693077086186766901318976\ 689140604292812127764585901945280036890142899973749956444785583193290233\ 437436023289637232140347812210493890847690583882449362732982205654006578\ 850392018497762880804241213239261171027665955567070539221124327451912749\ 107345161308513991671670630049486504592739249012049973160447933830616147\ 962529778236176003587768525585624463479676133018837864732273442122685662\ 101495323142604411319174288414929298505216726210544609796099094382261688\ 002599182015382193247583499289114645676398054409785108501896751139455016\ 784439940827059062785329667253430904400765003978072765622922615877053325\ 542063207495002690400682913340502238772623286846063916751732478593099563\ 483970987019139960276796130764190775599257805672838984798621482225863091\ 682035181542271520303615797945344306307795092562452439790916870985630694\ 831235864667110549555357834486979321905213330860465152880955797480142865\ 749778484871343926257538937944730212407983360545144093813544270153510246\ 493954804327772028845338918218454036536369485465760534992001907739226272\ 770910204005325695594921501662906502851930649356975577979564044060571152\ 890097615624228267745963813394307581340141230130816493795965266165043054\ 051245593423629332027209836031506487790820399601182802630070000852979958\ 661965178900865255629147999741637603479371592614226325089440035031427383\ 238095846538083673958228692674673302936889957441190757176639944772347848\ 776360677460801802021723699022088251953275449082047147781181486900036212\ 920485587140285897616731375528695422725289206247669652471328703697220600\ 422182342871615010254554137005973158666574177689186528822640855305968819\ 527743358166087695242161943033945393817100712418083120979823642139838165\ 289486145745425135448984937220172478996558003953976810284325656306941294\ 633910358071974939152985088158971711310893609935852320998162366994864612\ 034249728888903250793156569195867581277775834160831797366736202802025755\ 283514262268986816809799363583143809938135850240419681863297551120413532\ 625836518692003371774786759423177815394555686402435582674551315197560530\ 931972615413719099440402956856157794307899090803921954249846207762718385\ 856736086186510153884406826059545430990809661461349170442092387688671994\ 058393411926784102500644421414044482105974480282247551449511425723437915\ 554877311277561697464946506586270700118958315609245690798251580626547616\ 150376120987210280200536481420983867881383843901801170856261906801357803\ 760742658671452165202282214633796224129449081039543926062718040229619558\ 088766462146645570134661522618492147056111528534992808966909158276395962\ 882851902349427355796244515085051824209305813862155887428604156460206844\ 711777077254004863183951753434732138057484078095300358647433763027905845\ 948718316601203242614913755210481596057016800507396094899039598286643492\ 954138116563185939866979756235838016004805049809936795781400407662285181\ 251514711618754078706317092708766256640188925484283571507083597514702237\ -612 36811317086080345233045226430240154022233186744900 10 The smallest empirical delta from, 100, to , 200, is 0.03005874928 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |------------------| | 1/2 | \ -192 + 136 2 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |2 (10 + 4 6 )| |------------------| | 1/2 | \ -192 + 136 2 / B(n) d(2 n) But , B1(n) = -----------, hence n 2 1/2 n B1(n), is of the order , (2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(2) That in floating-point is, 0.020139539 It follows that an irrationality measure for c is 1/2 2 ln(17 + 12 2 ) - ------------------------------ 1/2 2 ln(2) - ln(17 + 12 2 ) + 2 that equals, approximately 50.65356953 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 5, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 9 that happens to be equal to, 3 arctan(1/3), alias, 0.96525166318992658020 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(19 + 6 10 ) ----------------------------, that equals, 7.7073575485293709656 1/2 ln(19 + 6 10 ) - 2 - ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 9 / 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) 2 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 2 18 (4 n - 3) (38 n - 57 n + 12) E(n - 1) E(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 81 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 126 c - 243/2 and in Maple format E(n) = 18*(4*n-3)*(38*n^2-57*n+12)/(4*n-5)/(2*n-1)/n*E(n-1)-81*(4*n-1)*(2*n-3)* (n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 126*c-243/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 18 (4 n - 3) (38 n - 57 n + 12) B(n - 1) B(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 81 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 126 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 18*(4*n-3)*(38*n^2-57*n+12)/(4*n-5)/(2*n-1)/n*B(n-1)-81*(4*n-1)*(2*n-3)* (n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 126 and 2 18 (4 n - 3) (38 n - 57 n + 12) A(n - 1) A(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 81 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - --------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -243/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 18*(4*n-3)*(38*n^2-57*n+12)/(4*n-5)/(2*n-1)/n*A(n-1)-81*(4*n-1)*(2*n-3)* (n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -243/2 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 4079037171087071295778036201075674546890919159902591971638938532266868812670\ 534713400017558079365079386478553215451646533917269178264413817267854809\ 898387429733104129862101184295082317298234040012128650199779920121849068\ 438591994881796374167193077474594275690154821585850593327201392550602710\ 660407574297517921809871511407060437701201795219351933469755867639728924\ 140326756428571279806065867624488378924923768225217386213018289574710151\ 891797552512469812893997499276386507647953652985078991174920551026905765\ / 5627019917269458164925945840984586 / 422587945366167627394061528355108\ / 359581210829026569232937489475918306362611663725693355272656208779480811\ 737258226004473435943706391576909687150036299257180721233342823730879170\ 997825928564390187268071441977717114826029950536883954082792132656048544\ 384931882101687081046471815992546658721042508916813906933463372916498260\ 399807559250505253366522966548225660617742135889615376799487494694189528\ 176722095150655958838553647257545397923108620685568047133935404730540668\ 480517319483389887632669143752793435497464868135511476928632688803051546\ 28875 and its differene from c is 0.24618544696340339021799149820965288797501120542659391337264512405222178317\ 571110624425890153364325961873068973358597526701117291909825117958547974\ 206412447112669393819235852490845992096932153806981479220938206634252019\ 072270879367067786891109253841514850126739926622347380263916792172257143\ 571644492936827825560658434461285417973312063304732930171901784584563079\ 509771047328909753768345639031740259614597825052910611737675667417571274\ 566540897867973016467516081301857756763960726491233498441145746598727447\ 436240279581031814111996997464604466277184395461110166275915715144878402\ 695463432320595267090296913305618134280085103847249529111089903052570266\ 091727068924783348821008518678272185955840608373460031240195447627567141\ 457146314471721346339285944698720501627047387559486127670102906425326300\ 410394857162184784988833689133472429112534204296844918900898434802444024\ 065717508099439123886896191041668868773917824034909037100462939001860829\ 271340770923132626104183620327708696625121186778481388272185752629777778\ 514556911095778028623152304287485877313285000919925719631142184484250756\ 682252372020125797814121734884686563073591957949750348000333512842569543\ 910224230212883855337390590836557111726537311146078176772133859437870273\ 676929531862596663067218996890011853801592520877972820618079272862885922\ 170251758285462605748028116328167573477729811026101073998095102344718118\ 026992885437079251730513687923850934308819383404261572330963381607533349\ 951533171771351735901883141134559329191722179454344186959116043875623728\ 269884950067938525627489083915300729632389175273995245478559957986598629\ 695622179391650474340376307952907976297679231313819604223882551285166734\ 019908834945155644483121356513287802484506498479075375817238209952707361\ 117972768050372937667975133000979970469755782063436665909543066056418865\ 354846734539137349418226358992235917226535775363371089189111733909057153\ 218548446475360309421744410079327836941141517579686006047987048918477100\ 230583346881550678068815474965734022625085132401258621847546630388500555\ 446924705398775718643169272688378182955455833271514645005982720374750369\ 824517335719379612333889471686765885098519136097089075562642406820992665\ 461600134086619030724898403106574026803036442377740814590572724302710454\ 322933020297200570394126066835445612344335823296932984340681096909614743\ 655060550508337361857192628651315891465966282480128877561424030174413706\ 160859795763478658998258994405836853148137710684931909900854055334772098\ 538065655884907661889437515938510635668471355403502453057288597009604610\ 070450925701730215444631434241551664581229449809219867876539219466318549\ 507776241876103099155583728677638502934165329583436940183121358381340965\ 073243950338575693985423526297914124675652799345328986459031775820863930\ 594117900095298430353026164161739844420575413092881114516130035225516365\ 652632584547039829062335018530147767997384409070825204602394105871606983\ 652972905967904855428703603866747921157177144731619525394462458125060746\ 597861905009025917831225839433146202579536554373273253617344742003366274\ 775172372968879218126406615003326354929348991831191188889234448940091198\ 843337744010301289774037179659737144708197437529377317789204294355501760\ 931624175171912904942320597130906022846217619264737794944149600692388078\ 679326972879277801583744920473606836377509565570861092895696719286250402\ 005648084212817130953240271803743081542914556153342580285406057390070964\ 801643363554535582432692749827517271147723426566533960247528281426681305\ 617742179113608579376961079474091391381632380763331091151080211771265347\ 732176041421902199883754388597459234164333676824832832583694847540315396\ 600461712180497110241890891122348028004248643886595894113141742622027617\ 625796108838154241209524568088954580218717278705708183823148831439126498\ 082084205369585586633205559046170680112745674714085280299886097532936678\ 325645025841564567175171546696492966429220650812701802811084097507332737\ 540730272895807313524669916305970278208282630567564828434546368563416381\ 336201609160055601845938917240529719138593309937419538486891737222761712\ 731244588316105009323558628108189738903099429740617579080955240865842153\ 726274067366741953473020740984863220378083203944067180384465215019527931\ 474823013419161806719408547835466377897884871208275639478301672319961738\ 501303621127819310991887801757154820886580726108088976235031550994138605\ 844033332451624086597894131286632713694650828440517342501681689800510094\ 731656498643267976677916938482446260542427299313402656968611364929344917\ 805614621169378868960701010833538356490369668548346456008027523865397082\ 767260461727690660863575662463996916600756381939836054429632211760954116\ 521522115514887258023195931896289551344692779404942048891072316984798742\ 014774319210570924418145344690239644718349725940915342080381799964174190\ 047467885560007281149111253571662101810129394577469395636890140759341713\ 122960030139044258728789009848433408545236226240196489013215232807397861\ 235283129675544605887191451763667583466540296309537498582073168586189922\ 712930948250198661552460994451417181916405297004373484103785280627770636\ 342381594950893365993153510510106077695950062519227977550866891081424486\ 655731204883366059602637119993027930217269675670476391831817121269580977\ 974372266477870088494413013958456617298142111877869030176012884699587026\ 486596203555895897265213995342695758576942745653136403357565114389123983\ 468640832132183827531503030040827744241679190012013552626691381669356521\ 635262977442442136092754822094152607059947426970039276580019268639892277\ 953399738049306298509564292381678050011645746211043146290066514403613724\ 341688850967945108887160809551067126387811650244103687782579574531587142\ 530390908772360442785526523896846860656666611989727725982028483909479444\ 331413006198762560671887132950281434587782012738921173319235018317359760\ 621373819122916241292828646618314015672176185051221772720646466772181474\ 174708108306923412936583926916163784627417598021496966785477782350938884\ 357683743584212493945949759304433955957185117461991645463299300011850621\ 169930348704589852514397361022244690080156406032324667494967365618318641\ 483024996095522744003369730859906379585185907808182677041599998393126745\ 360038350076206475659442357220605129481187841896948103282327136612531396\ 471279900782612213856166307255515264837247823875755698150226865304818425\ 451391833674974664554852255201552452991606302509649660086887719425186674\ -631 4621408623594269779175120443899 10 The smallest empirical delta from, 100, to , 200, is 0.1640478788 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -540 + 171 10 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |10 (10 + 4 6 )| |-------------------| | 1/2 | \ -540 + 171 10 / trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 9 1/2 n B1(n), is of the order , (9/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(9/2) That in floating-point is, 0.149090007 It follows that an irrationality measure for c is 1/2 2 ln(19 + 6 10 ) ---------------------------- 1/2 ln(19 + 6 10 ) - 2 - ln(2) that equals, approximately 7.707357657 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 6, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 10 1/2 1/2 2 10 that happens to be equal to, 1/2 10 arctan(-------), alias, 9 0.96853408234038924935 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(21 + 2 110 ) - -------------------------------, that equals, 21.305691483920764813 1/2 2 ln(2) - ln(21 + 2 110 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 10 / 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) 2 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 2 5 (4 n - 3) (168 n - 252 n + 53) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 100 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 155 c - 150 and in Maple format E(n) = 5*(4*n-3)*(168*n^2-252*n+53)/(4*n-5)/(2*n-1)/n*E(n-1)-100*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 155*c-150 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 5 (4 n - 3) (168 n - 252 n + 53) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 100 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 155 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 5*(4*n-3)*(168*n^2-252*n+53)/(4*n-5)/(2*n-1)/n*B(n-1)-100*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 155 and 2 5 (4 n - 3) (168 n - 252 n + 53) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 100 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -150 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 5*(4*n-3)*(168*n^2-252*n+53)/(4*n-5)/(2*n-1)/n*A(n-1)-100*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -150 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 1899226933473554819977923304628361190398882731544073002383124971732402569371\ 542075724821131121022725635237698815196565193904165508400494401924647568\ 529364758456484898532018468614325507594276918515875095503807216499961919\ 936091138390663161583673830204854175975653642606064826620231090318022157\ 638887082126655622085951013723857104881749957311747660517595489307428014\ 539608169724981167753342321167808369296566652606684963494263368821926295\ 239158167395574272729854105942049014844791007038702484421942089839117683\ 457789537369223888444478372056689681438316491327774099069779802372902852\ / 88810419215236556000062112874 / 19609293757471257905402455896582008013\ / 570863187829714661938526885370126041909701911407811614927372479871438095\ 652599600544611619008999586434511785035519121962967214270527286099009031\ 851431898759952709672801561793099838165838303411674535361172979181620869\ 171565517481397707646104901407737897974650613176741565235836340316852809\ 526695340690147654410376439677346095415062741384748422339320157817618519\ 413013316330821715235899931985301735839033311296738204403341141779973376\ 904489031404905564528107086417028007445020419133749409126037275034636307\ 4167694031130394423304606829379905657198890573853203544913014851185 and its differene from c is 0.96435539527762632447974043673732130720593606096278694854650810666910886774\ 203648208112322289756209180741867137688558196357273928031288301832383811\ 403031237160578162559422775755248523461313964084750189428614827642803699\ 318892879644265472794449472202110498267159092098911457857354282117764145\ 898143898298582301189771170190564996317360905616745799044390278392934987\ 146463809566910368465243975565634309112881215714523925170613253141400824\ 654357467322531736376814756627598713382582097087096008006057524915865002\ 830506752737917575308946978124542644547108837893897113487205182269054810\ 482252370649393808948179571249606434353337893214044694180344998070371475\ 892129206680110802986153554236868317097268470155792088851292849571627168\ 012861766061496207441224048719536549373458876381123596549497376140121314\ 060375126466793641436116667673708698917807539325085950635938827932960277\ 774322114128720863618870120402175656796337740291822062827785596972748086\ 553954337911063364575481441573130777124255999696593695479664171243019081\ 806658139818324205062998967764097185393567618312977283930958520348085111\ 415079839260658704629662585630256444750308410632022275603353155793587607\ 013946313918040914447826283637033018818564271033378810221949704439259224\ 384458744456269174191953420786365411505556620387368611234050844084377901\ 125370499559917971788232109702495246915202990625387985535527753437885490\ 568039764205730347406472562266490495684784849681031129055945189076869043\ 599186276751917902620838652589681330881817495916973490583431019925832500\ 282440541950215764160182716368271606593739481530098861655311555283457123\ 733532100315745876941088499976436865467595994728254060597000764639582439\ 091567406543647553264368362566647176325602898756967451951689318320476758\ 726495971693161788468827044842122280499377210360730681258866483536064212\ 360609536291948913160338660361653492447881364414819075784349503302649358\ 320445425162114729144964625730464312436084556012501959311655701984862861\ 024463316284691583539739534643459296751729732033090979434459423818137090\ 818240781012697939021649304825931736954518457607832650973358222797118289\ 312581753050073951010574437005084765315660237523838703104111254789135259\ 454104103402654722746492169108742902189374565690482766747085628069250814\ 639780159454548503239074406837129918092099959297591722017550739762005518\ 707245651846031909291845453720472292749952000544634551799425001763526113\ 092487391048652986926649131361721601775306099056500148545371807199196262\ 856117118332785235171188359838159789414004190326213611664409347696769240\ 297194011583281454810958384055250073435147147902739689293136420652680566\ 252148241104249924562478042211392737367047587214840248603924520611587544\ 325214754935025819203151070173379498581399762071904633078997342048796867\ 129747789713194348497618221184117197486746257595971104707911867250621900\ 657915150790250823897454396388016842082009438561258725220042761960198833\ 007235447269953607730874175446936248785763707723461115415389498506591459\ 844641873477045756336744007191397345943338401544288093917582834821964283\ 142862775302058913314675924803076769954299707060993505655780800238289040\ 532952949828878758394616687645508946371540656016742695686945261083442653\ 842293028971832803457474590701891855132392802598527726859617203712372505\ 783859549284749379433272869657035083003409158887863021748362453292696664\ 471337210684510923997019083160279774550535445367834649762388061910261517\ 763854472597526948015524206041618456862059310087276037321836597610761507\ 652771096221463207861133010439963958955553422892690122542363966414025333\ 307806963664970118382587176558153318133709073642493626334796928291714141\ 691312345347135240933704768733334796137305308940676811024804170744186101\ 842385158721376544656753206671270289768402620530898140210822501550625457\ 650507836319085724199911334085728864116401935330340268369375798198993068\ 443935747684717231261987802737091098064696402778358260490446127733873313\ 117680623631009098912134300349386185167648502667389427745384317667095996\ 636423696273257877944321917922395185055944668962425411679764637087063342\ 156828759889611191615309166853316367047206487125900529613348274635257774\ 776681562938435821324058899081770678748566586525638672300573337978745570\ 563334359169698863534662291475082440540544123506272378358705015542350784\ 008935156496258802801900661718568385425049474602924577584634112412984516\ 960186705858277507165510439153557510841043526673759561355806455739659112\ 892614457490255295710200207079397240249323258489137290695424220851335594\ 648816480926523398058638372908427226098957561606051828667283137075863781\ 216930219880269686963153067874311051466625400413234260414802242442367452\ 213891025894773178048354435372475670921409047655111287680265764506412380\ 670214193952277901891579867193402127099330226525957394320844084718641810\ 336252082303168433693467385247806636041061474638630891553630383071663119\ 161296826399899993331136468226654473508177727276970780011526099163072976\ 281086616573113833055254722844500792852153038458836495053061165966154883\ 574117379607929416522216602328666402723364815712583460445726970690493509\ 311068114178258289979585265024966007015594492872477836197488472341040967\ 542275006100269804026861153925709460147646349577807579972952048313322759\ 308739757831337562461943733561562614145158403658749972776820949536818850\ 844838847975047732283915210032976468950194838203697689219023097309099126\ 139859557371054569651700947161068110936241665543399367004897133074011980\ 617837445817552254947767371295492971132708653649688494421414379479146812\ 599686192465214386457687331638766594581790102847830738102038548853737160\ 878119862929483791201877750102033585599941881140449767696621270605256372\ 166707919443748313780625961187242133023323710866592793976879327763419493\ 360824220973389700391571911992796935083036973382883426818869441667152940\ 659671114713466601137241556256588670551868102358812076031000033452082474\ 594478106912844554413460241718728397055294079593389917223410284701268563\ 911206661566603270607939638294044076184314401313203927827841993829880608\ 981241022753665380903468213261383445107570757297865387767842090624075905\ 210194180148280148374832570274611446651386502809562037564822610292779029\ 586410428040314574227503502392288774889855821138323395029678245592117590\ 132341375889918546552097185454285035388547640922368180906737342161051558\ 402004772792842132943600384066942184769668572576522350030484736787324501\ -649 5134078435614 10 The smallest empirical delta from, 100, to , 200, is 0.06089661646 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 10 (-10 + 110 ) (-11 + 110 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 110 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 10 (-10 + 110 ) (-11 + 110 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 5 1/2 n B1(n), is of the order , (5/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(5/2) That in floating-point is, 0.049247276 It follows that an irrationality measure for c is 1/2 2 ln(21 + 2 110 ) - ------------------------------- 1/2 2 ln(2) - ln(21 + 2 110 ) + 2 that equals, approximately 21.30569163 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 7, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 11 1/2 1/2 11 that happens to be equal to, 1/2 11 arctan(-----), alias, 5 0.97124959639138086460 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(23 + 4 33 ) --------------------, that equals, 4.1879821595890442720 1/2 ln(23 + 4 33 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 11 / 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) 2 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 2 11 (4 n - 3) (92 n - 138 n + 29) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 121 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 187 c - 363/2 and in Maple format E(n) = 11*(4*n-3)*(92*n^2-138*n+29)/(4*n-5)/(2*n-1)/n*E(n-1)-121*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 187*c-363/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 11 (4 n - 3) (92 n - 138 n + 29) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 121 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 187 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 11*(4*n-3)*(92*n^2-138*n+29)/(4*n-5)/(2*n-1)/n*B(n-1)-121*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 187 and 2 11 (4 n - 3) (92 n - 138 n + 29) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 121 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -363/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 11*(4*n-3)*(92*n^2-138*n+29)/(4*n-5)/(2*n-1)/n*A(n-1)-121*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -363/2 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 5929712003858438675918383761662036271290830913054447697164362989256985085285\ 929272792877063102206512192125245507515270065557793490942295070873566029\ 484325259386753737066245056889473491386760175730613175076275283913669787\ 511938641074530639213829545164935302097294535703215463103638973994624946\ 884988378345634747450597699385109392547165726606088585250680701390801540\ 687060747958605626094193413438449151147936228163610057947164412667416578\ / 8220570418400948014567324199785197206008958233896135564102168687 / 610\ / 524012147847989276358238418715841669787083419025506654823113515234521334\ 972922734548376561587291370159643868895565441794816984235019565754455815\ 126524036137393385623491213522572306974766581697015999206610296022060858\ 216319189559410573381416218061469421419304204025852611107940289084708872\ 631995968751359502496244697287497808874064849606246949786653560458413023\ 084453654058074160982196066457610088587322997159770512999979820270834175\ 14062433378276352535355421636894023282288715472372007273342944000 and its differene from c is 0.14632996937262071547099126309458102957758786014011738441892676013460166697\ 046605091329822342607372893631334233317050579920131075590881558027408446\ 689308220402740745133204516091422776745225931721594205724814311821304152\ 477649653905662983225859810722452444078253410775097907195783602858001710\ 601552233955407653459711220106421465822604173117794577482383328763409058\ 320523197329857902489213108138676553748773593202954563667127617680137838\ 814148578294832047308989998988102990036572618437867529820095405406695674\ 610911613511879274069071260547670408688743582456925486354015172431804564\ 403827499801326990314893128607131223750392667814879467832990670163321269\ 726017658171260154312457624916069267484871582207504204808072420800280516\ 972674246695027110539676568731039834493810133099089573510240520983931855\ 316040697248283870300172757327646904021153127930150005798947390028983743\ 655935667452294013108076737915190563329304445277456886463936089297470725\ 250208927912055738246638512277909476148939302731998683881213508591515671\ 036502025298433241799191837678085928443941720338373036500109180182470269\ 761733466092041861213299023013450987630490321599783287646745216101497412\ 629642622223840057087415395815803981482426422805505483653032841713848308\ 320162785059839769846318504289527031380237902529746372282793315359453570\ 297340283963686491167821508418831891822447252386066648011325740468078347\ 664019131431212794405246134591721374254474913458454997507308910281819631\ 846703352148605544280703889088741305388346605737755385644486309077271685\ 834045108713131841938988649575286453679695863593961070436508622704779115\ 967299175818040530880772192335018749622253388472790814755896795337485073\ 982787284321074277069352487864325609933541230062800704019873089135292190\ 412146498403398857923932004209191571708529548612793904062399265872173028\ 850738665658088669662348903262105933777409131919380232832638335252628110\ 706630467236032889095527583614315577638289110843300952751384156104035674\ 181882885248299003691007803793484582497921280284741588638889755815109070\ 808792282431414926366838051391762441315371731174054846961989308675707042\ 310822290509545707719686016032657784745069317326631340365416401729010211\ 852796439650363709283684919765900885425950750524970690098580927170442756\ 376028598306249826651709664966629729921559978413624142930691441805375921\ 432326561726700340996911102201704352061262136838830473756272712057144363\ 960899657658509760578632122160329534213770173956450322185237567164438943\ 180755113634718778288698630851739094122463984606132101294615607849768547\ 984753226656512006215760900559232741015025490200689587673409403316699126\ 331754328685907195897501967806034363513765072925247349341495655805684476\ 296985112583566258791585396759722279499758927512179004287848168242059878\ 361872563903326399984670451187070498937053721715239989012523068369233422\ 727503412936297078939933558225271307661403433538673121824191731015971554\ 221537297428159268349913486604586390438259230237451452824763862632176143\ 566823078014991937913720762723810686202360091365709552057781455701264306\ 656712855261405571891918417729286401121573774614999803991488952625824577\ 394292082845498531502704976241658686096792244678433998390461331215575906\ 749483706635949049532944401959066380742503910699953392151437708886113084\ 976636126641365198302947503325722823207731623110699059792694666235029972\ 955183916005948374846783404111683045065759667887172841887904350327472964\ 797682790565987440402177330882792326739053399908281957845400058075751632\ 242392762263897048032565342480978403197280404166990518980025668733840831\ 014516040265366557288529535017875265656390267862484276829219441369059412\ 371177599461094491856763446302315022028272053048511833560236318738670517\ 689141745969422877902231769861349088145684975804989658362451291077552477\ 623496188217334514416179960497335584035147212164151380197425719987992266\ 577167131768575169416810706737801753472871379908246510279214039246550930\ 985587588502970973089592896401154107525025891116834338571776667936360349\ 109286084221928279008000065605718338039966852750425277615870116864048175\ 384459518033235436755140263534710396383480214213732598869410782365693076\ 183278584690872534209973369910340533716271530157579158715255143974113193\ 795391755641519924690758977273505045467095017778793423090406342845721049\ 954411807510028528487081020775903243484332933991674127385317847946015553\ 212791028860088009481182107496103458586877132439985401479024560624920113\ 077942176274543402565022449924271156990619538356406024621580937722482068\ 919938800297614991199687011452392747544648939876874145073165359637178692\ 534502996845872890874115731839589374387283650696477977982322320029169678\ 238252867086237426233825655167415237915560881932663665599762916963658890\ 765291286013808504575798748177425057474279326553090043460623111983672111\ 512931294751137533376390810957933574579643335413547638190407882293795961\ 755211448691385200662813015754783197542431655990419910670390774654921708\ 373466052562466505272050793643872480344369574017325003555788859294126628\ 559199449049658567506913016087232081023208171743833442128845663396455950\ 527967187388911882101309194872992103910240556618440997024192614490849302\ 079224910745376815508782903778929201042594024203719552064000361634501013\ 957723004693762035130508657882240338796401226165262404289961021186065441\ 213806155822274322215046213596189033365585058940642244189643248984738051\ 107753395122409521080003969960630414322497209384406869739083454425758075\ 314971335594285295983966682989040358908175372367790086113837108953274889\ 536563225388661023550608797494813916730191282727872481259320444792687317\ 832138884433285248274151164675582983682337246568133906682229198177363170\ 690897481880399080351331023836438279098802289183442365402210043491258989\ 754650973365806722559458300649334822174495504781744331865831789221418060\ 958729092976097552069095400956723942661678396466426960003138187024489126\ 659193502653843380549636079405875471637685291357494246030256759756086930\ 211200459797054624911438460647996308698647422390900127492409360854080924\ 237775598890664894786465685504556303574170404992994595563980814453409170\ 906775634489747152289859923951583369897204629227321995711999561503932560\ 980697973578967015559263474958507221257861060215113402927123885393101596\ 769948900706743033453736041455409034275299431111541350992901387191301094\ 8773855911140803274666937825818033895479885677277297397393901140746767 -664 10 The smallest empirical delta from, 100, to , 200, is 0.3204553278 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -1452 + 253 33 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |33 (10 + 4 6 )| |-------------------| | 1/2 | \ -1452 + 253 33 / B(n) d(2 n) But , B1(n) = -----------, hence n 11 1/2 n B1(n), is of the order , (11 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(11) That in floating-point is, 0.313677988 It follows that an irrationality measure for c is 1/2 2 ln(23 + 4 33 ) -------------------- 1/2 ln(23 + 4 33 ) - 2 that equals, approximately 4.187982703 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 8, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 12 1/2 1/2 3 that happens to be equal to, 2 3 arctan(----), alias, 6 0.97353345620597635770 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(25 + 4 39 ) --------------------------------------, that equals, 14.892104575824394552 1/2 -ln(12) + ln(25 + 4 39 ) - 2 + ln(3) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 12 / 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) 2 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 2 6 (4 n - 3) (200 n - 300 n + 63) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 144 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 222 c - 216 and in Maple format E(n) = 6*(4*n-3)*(200*n^2-300*n+63)/(4*n-5)/(2*n-1)/n*E(n-1)-144*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 222*c-216 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 6 (4 n - 3) (200 n - 300 n + 63) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 144 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 222 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 6*(4*n-3)*(200*n^2-300*n+63)/(4*n-5)/(2*n-1)/n*B(n-1)-144*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 222 and 2 6 (4 n - 3) (200 n - 300 n + 63) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 144 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -216 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 6*(4*n-3)*(200*n^2-300*n+63)/(4*n-5)/(2*n-1)/n*A(n-1)-144*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -216 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 1450320155431694490365553335825449802892540420719459623544223197566019370169\ 789627221592146188204675002334300715104369325623908728295062393022939619\ 460118522392331533951040947365405596969432576283933131003531131901788456\ 144405692082666557636178577044905556518978415980833994363006208042993865\ 685024809166248309667533590488487129224570233544406227481933245826435794\ 064666085561527317654301693283658501102244946376235470169882894046512175\ 834199435899777728678375896642977224623426992479096793214905216053205069\ 232558027180566746022286275747167739854899443062791820407754411193964160\ / 85462042916144948076871912453839891857247809764 / 14897486534092378319\ / 172530509569658747934903621206214236810035404123394621897743931370592430\ 625456526456675790083794083723056610046673657954483659391916670531676833\ 065082136215071513591062464831054227615587260582898518530937311521246413\ 078103932314677884709384399650960700545957897192570001172917969782622202\ 708262880255835846131598199456920993129233441452748986269888013093167028\ 362457394501780251983195538339733800922739244172684825341649243170683684\ 437015862113075882992853760115320221678942769594899784586072863354358514\ 057233203989666725739677929488584635929037234044016284109600005430314318\ 0797911543187300713703216982625 and its differene from c is 0.46623949209030836993668816837327740029875282177686158179229653338636152321\ 151822313703348267176823670033367142198556044013823686785766215300635360\ 600685781064961687354138557256107059847750860514583260763285752892731345\ 447427123940461529677625925662445280932430782626381782743647767458626459\ 667118040399451598921473886874220432103577441876991679026488814607758762\ 780525578796099256059213995383522763050089679022674657226987876434869823\ 473102547599352234537527673912159783680313735865126620305693700838910103\ 259324513050675643481844919738635302892725773178022013625960486327213247\ 606501060212546970777064567236261023655148459417332515336302702729317891\ 765135656012283868344531048721856757438181972742341898791834401148979811\ 803228241081124297563656992844074632523110828164994248177561288936576912\ 640676169409021428469432065286988642459728703260555000894067030972317185\ 411426954622175395417770100892128983265786259346614423068368340399904547\ 543920131278042931180928418595476360064168867562149476777100735493127064\ 010941360194375132335490003408080622944897237096109809834555398176345307\ 300736380673572945857943631485695382597124008691816064409346336738696871\ 577422656470872278971340443556578647373532247006690234202461925198920119\ 585739746177559673410321652651885601729219616961960580310757783069119271\ 095464047973460047018722697685417143953129035376111329422257718721265917\ 230852144316489592786244226559031307019087999946137780759757499221394964\ 926745693761481633757861873253409048684049927333541353139257901177558876\ 490994131754655102585513512278790795987502052569018803904080393422369505\ 891374024070872753364906775182438340021436963496986389158995724641057846\ 057768254802973092868258039775562697284335765403588627601791439812062109\ 826272646639225018648344569664052860037698686295409997224345904458360262\ 821454366263531630749976197050399392915984474237753312241191507910168163\ 521856366805823870655277370592677993840443928574411911261155761347598354\ 378746586430156290867608402558929161818855959468281612643176998887295923\ 204372101194175743114326465879030115143330935108767562907042653413948613\ 612526270839684431832475709317133979546755563504034981413620870971161404\ 998160524992941600447692655097678320115384827400159772180548997798263391\ 726038568231403214767955183673900020400848069443080991238335427633355308\ 359640855636063160160431731246647743907321963499898759575085105767191909\ 126145808092323342623730794954007171454350935430733523064898191837941210\ 114417126837600083447719328982425582617071555464126993091325981755919483\ 598444723797877428492358230364412657869950840258325373505210163337286654\ 419559957150422208647109644243364793487188977716570194494979739375975126\ 707184080260610407881417128679876151174643453729743734853055847634358393\ 948667215854699026937102723976543502797873435239626762322667859180680622\ 188598058610446988601892200446225851035037658451712436268235604065638124\ 022039555891512324679472480128566214043481757681669751163398479625431087\ 908503259238314633106931520672546306824572326805651468022131153038481131\ 921715498602230676501079172534476060128519504460050399204997033991910011\ 125514639254223435389891138798877115824324694153155248551686171834899876\ 413003484728306227279593955607341326674079388577878938379012581029381812\ 037406493465144964524081374085147805897336681439965885318893393491055518\ 637874355595970407066430197467213866808646474646458614106218801502204431\ 145397940044319528179742689882396118663485985250706871345728591026530482\ 001026959832427876242313692948040343609052468739035738328103733398422563\ 999784919280608868277929021611275226499151211878287364524064599190376230\ 707774497188014585742276574304428668552169657656008814428052334458061917\ 708388325971889650552228642222403120717243021830647949002937290646116797\ 150810642928740151414581678207777188544896953874203439559549555157876636\ 992082316018416049598128971036383642416205849826839518057505680352707102\ 038339237076666565773599490096629829962760859302525945728525473993928569\ 093737090650747756484031174994947262378795704328160553944656646957675081\ 662659562246212274360070930980639786558815670771354142386243317678493889\ 607992469025270961369790626348890030668911365064552206474874293044166027\ 542320040488139195043928598434723704237697116358287885331906276984901484\ 364914758106351989491911285169572213354587142910281122113924839783249554\ 049703430093001165686502460165383500293367234427531231146978826113809982\ 240926192766287271034032372223772509625073874193090237101847746822522360\ 999228165426738819213476306604164587205782437635611599602365221406677883\ 577566008031372881156591318398380121103921134520650782072236967763299331\ 165693451884618190113063415432674270413768326131070143599861168536476696\ 579495190353866839291748395278406717101054169209014293710220618658833311\ 635549965144628732207828825868748132539790218086347792944139014655743515\ 000206902006122959063157860818858228739796766639749274386364699399963689\ 139955151323835610101375384884082273048088612234359586230388300729859826\ 200387314095916389868942632675928843031178914777664185712867453639047827\ 035681675568553713276520904427469346669575601918086647435677745747229061\ 287507864471089655812303006099767518890664462055722861766919294001747846\ 236716081828223224317879927772044757581320432759428815460051791206598449\ 277600982539739214595914537967624429624215421851508363689851974816999781\ 548162881112233284536461548919491178115424737698019632182649062235578675\ 576176466017843052390171392783058901739628562061245997133755623975892248\ 621198062853681122830335507012178070427370859876134405859401897466582811\ 680106535900726180435875120566084007546357787613566326183225279419452697\ 324876623214278585453310983728925733125552803615651323108020634467464060\ 343880534735764195161548175427803559733405302976748200716313437548680822\ 060744700701021270612715477887711758044482770856760193446268882298228174\ 145914348862721091917358798283382183291854524392888355367041659771655010\ 911011034815512588602102754031542782619917600875264668886733293758789819\ 973580295750734342513451479143561013447106333507484506776250503586259086\ 716198647785709229782041671802939719655566825218149842814705820709684628\ 969626029879880942313765371291617271520682061856621353433948906145153168\ 162028797709525873174760425845041071829417442021784446027126317588074031\ -679 9880233100203567691797190336783969649245427281155673848 10 The smallest empirical delta from, 100, to , 200, is 0.08444864228 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |-------------------| | 1/2 | \ -1872 + 300 39 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |39 (10 + 4 6 )| |-------------------| | 1/2 | \ -1872 + 300 39 / B(n) d(2 n) But , B1(n) = -----------, hence n 3 1/2 n B1(n), is of the order , (3 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(3) That in floating-point is, 0.071983381 It follows that an irrationality measure for c is 1/2 2 ln(25 + 4 39 ) -------------------------------------- 1/2 -ln(12) + ln(25 + 4 39 ) - 2 + ln(3) that equals, approximately 14.89209545 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 9, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 13 1/2 1/2 13 that happens to be equal to, 1/2 13 arctan(-----), alias, 6 0.97548104298387346570 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(27 + 2 182 ) -----------------------------, that equals, 6.1577154229894323965 1/2 ln(27 + 2 182 ) - 2 - ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 13 / 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) 2 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 2 26 (4 n - 3) (54 n - 81 n + 17) E(n - 1) E(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 169 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 260 c - 507/2 and in Maple format E(n) = 26*(4*n-3)*(54*n^2-81*n+17)/(4*n-5)/(2*n-1)/n*E(n-1)-169*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 260*c-507/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 26 (4 n - 3) (54 n - 81 n + 17) B(n - 1) B(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 169 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 260 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 26*(4*n-3)*(54*n^2-81*n+17)/(4*n-5)/(2*n-1)/n*B(n-1)-169*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 260 and 2 26 (4 n - 3) (54 n - 81 n + 17) A(n - 1) A(n) = ----------------------------------------- (4 n - 5) (2 n - 1) n 169 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -507/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 26*(4*n-3)*(54*n^2-81*n+17)/(4*n-5)/(2*n-1)/n*A(n-1)-169*(4*n-1)*(2*n-3) *(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -507/2 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 2123051652690592924526238185848163903215293522860170843566694214572356548385\ 890654221365435745190524103637453194187496164169718621881282906719238009\ 744093678796547111086606984362464890438597292050109606732983439679518579\ 383530137513448781146766453171873626466948175684498418620280719357695292\ 428457030272894067507206040907825467920563642526654693920154086254639628\ 050779818836934317490640078427935219016104637794811546846174704753759796\ 724686188535880884909509975305572830725449351007865669625694470432299530\ / 28441646092465163268696516396090863770177114471692507599829430249 / 21\ / 764150804985873589853469897788812027783221884251297611077319861851160462\ 949281777949505194931835506648766573581959956712928670386345546780371883\ 899228939683805881048780252586146725388719467762094158771649791569572831\ 838489834409067779172780225603578109199449998614025583071446680586369635\ 924015387176289807800801840364344768123065634852038585158457513752382398\ 127155624510408354885179648146836292547290960351235448583966046685160837\ 720730191101035877419233347260687462899277689703114004473114595237595458\ 6715640742777184925735323724628433878933270498230053700394656187625 and its differene from c is 0.19491204734128700330504889624770331728662322655269700288491243041699328279\ 884439112350713360786745992851998209958888395632034886174574325890672864\ 729339337475731776019127855155332717254074658522136578652029920031797448\ 468644935386647644006113809098487343706013117001931637770608141398422448\ 186157799653323517180201989555524612140624267844497020028130083318554288\ 140825933181029837352200009122552596592160202518679676894219383635856546\ 244173955443942274248211309425423420839023656112518942116325773293781679\ 655100935351832669913330234555758325866648308203654394445492921450191180\ 992860108145002205866577572443485413142163320747529192854977929790451377\ 185312259198370297826082178680360488590358183700500454459583938877996836\ 729461878616791693212979655298883343733175140303815552457268757968943899\ 494965728135558160327115651618505575647828856949584600356005471712475348\ 343265601289313973666102970054068939559990907406521966753309167916415168\ 042233727482973118524725680770178743581436791905819018752934904288483036\ 873119302964867265401296267036475074592496971776287314083226953742220234\ 163036942421892671392796525666292825886364880224145218820161651744932199\ 285482753595179024859589679909499278376526440421986673710968056863626662\ 427784867794095097923594183555079844790339083417354347246348560426127940\ 834443595587064907121346020596782289517181038638527231042881341574261026\ 786959744904287107507402764836501623003689008613432627126930076613118342\ 501326033378662644754925202272641446542993691565736587697131049457435654\ 596536263218316570731448908573584549107651726854313450332464172425366186\ 786044230503893809466462794513455200237711209749696111569046204753233489\ 052667674250326341549765296617247140941554958505097087003161228411586783\ 361136036468771882071424520236194506575111638413455498319810752619772996\ 477181860928389074386319139768028220961170436569929646454483773187290341\ 081216529542497856370005283564299379797276866240684895903105385773815912\ 743852035184190126035612742234725066776277204678615327828772641765330148\ 632542702118311244606907047798975342868787457848664106858637910832027620\ 003250340312543934685354193581279912801909335234033863462400348517750311\ 988313399328036518809001807051741739956721863408657170913233878713936603\ 162424202260755867143063758506191968074316944324156022672634816411806078\ 481239537432101952038017213571498225047703021914281280745539278735821804\ 138502847360624412775163611560374375386266435671281412790143549031661112\ 598323925007635546917524335920205582431219581618584850347747692635187993\ 484047927857662977157220635381191330837616902203357429048859115137504542\ 575018502882675979161906236725636898480066990190881258687088309003070321\ 014245705653521072754881193412910186251619329166165417725879398598225574\ 858531403679012835480116660595668068549672123511596951930699606300833422\ 240641849479841719878243682318940248276196625792035403714601841847510404\ 562733199547593463303061998768345040293234313258709405620862503783474154\ 106520440183893576254984275207111865559074204524205316572191092694765195\ 439372490429872346261836609756315714819725571926715322431078997721424806\ 295328070028943709270232579586237232089249203492444677438958910656609480\ 752947637607996319336364825885517586826388660541750388093892551932019093\ 783551289147704979540806564436748063549296798845436254468430289406017736\ 220513421899016855496146183543310474783298138656680726914310670829775969\ 759062855593615428451544343631665486984823245506250622062448104633201730\ 926002887787232337087280034496156271339655255619242473392413999669372436\ 300233009764233707228915571273925711698282105948198909909851723220120410\ 290791139938053293581489778955591283852007675630240779490836285054312289\ 990129457796444118952705138414570021950459660892726817824065553435126589\ 644676555641663010697250635760950402326460089468539575195280139419227993\ 411826716956694020079107576107572768583027803943152843681686839440481926\ 440937301421576122736198090492063178395204008558106502631095356836673974\ 319307309498607652893151745304105284175386799693546250838941128630782516\ 392089089224871479183397796427094080426952877323088889410053206133085701\ 359129065356708251144846332664312594612198413248114998088351675292835383\ 544350197830070570924841823280659506086482473880428844492437639063741858\ 867888315462498595988915611460785919925156212264298817048942664563751976\ 634263027744271690016921073991457870651102248725975105230453218971413748\ 957897444062559074460288136925412181706215405510397781189626014032530991\ 305512671298949334718576750114225783614714385209893273319929642453335893\ 459613180012674468728659629783185924425601285953816948305502510389491256\ 931495141121670179961683498567932072247485446889609686161622731120671782\ 423190757928851897339106985376495874807953451309799646610559040003749597\ 751580938824958184999871492629362621813177893481684894144494247513734761\ 386041581283076291250566916757865612966004435582674432544297918000336114\ 010536405031291449223896879238873688683936462425679449257887669576222926\ 895819829563501816911829740906284257428164424600220639344836111076598825\ 494618786879185874372830779499340141874386265520005674055055208289030036\ 250715267157678236945274728820445033724397736166214325794686207642897863\ 488470661588910222613347429113335377621947000278247746135012338137326700\ 921527944341145313139018062295567615272494538376833432291785024958438885\ 496918679397798732737670982571364692564889149340532835080049138474039543\ 053608514721969265515641774358793425852899105452145694751057757865877048\ 716612958848733157831266565741975575094637841755250491177700509525561786\ 251038062940631476992786404517393894375982425120927118412555929967140993\ 660275830232168209799892380780238936509615149030736666767177711722407598\ 696257518929306328900127160909243503352148017123694020819860071321893859\ 672821253067649863900047979244043100620329735759965638137945844967613034\ 320882596448473672879423943670783499930294131349867782508730612205752513\ 088113859540312229125328235428234580106838661287904060119210136072202734\ 081712016121309007219209774074137308558224521255360306443884644544048595\ 889127991182198180806579211576577383563414359254555211360704684223466179\ 294520138740070306872068579736274766458268900833600947874039814280790989\ 213421481371776032422317267922825770451416102399810376863308419053976660\ -692 016073069475279889824731212145346757346760 10 The smallest empirical delta from, 100, to , 200, is 0.2092696867 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 13 (-13 + 182 ) (-14 + 182 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 182 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 13 (-13 + 182 ) (-14 + 182 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 13 1/2 n B1(n), is of the order , (13/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(13/2) That in floating-point is, 0.193884291 It follows that an irrationality measure for c is 1/2 2 ln(27 + 2 182 ) ----------------------------- 1/2 ln(27 + 2 182 ) - 2 - ln(2) that equals, approximately 6.157715433 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 10, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 14 1/2 1/2 14 that happens to be equal to, 14 arctan(-----), alias, 14 0.97716155561605837242 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(29 + 2 210 ) - -------------------------------, that equals, 12.050569329465964537 1/2 2 ln(2) - ln(29 + 2 210 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 14 / 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) 2 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 2 7 (4 n - 3) (232 n - 348 n + 73) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 196 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 301 c - 294 and in Maple format E(n) = 7*(4*n-3)*(232*n^2-348*n+73)/(4*n-5)/(2*n-1)/n*E(n-1)-196*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 301*c-294 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 7 (4 n - 3) (232 n - 348 n + 73) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 196 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 301 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 7*(4*n-3)*(232*n^2-348*n+73)/(4*n-5)/(2*n-1)/n*B(n-1)-196*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 301 and 2 7 (4 n - 3) (232 n - 348 n + 73) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 196 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -294 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 7*(4*n-3)*(232*n^2-348*n+73)/(4*n-5)/(2*n-1)/n*A(n-1)-196*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -294 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 1053551203347315740849307984035616304388589495900505430448754475115006283678\ 149423011481270125540330868831096303954031326326349939732819835481770999\ 756296261303438390242926894979673422935892687689461623436235537182170867\ 497488584219853683926318422729157703235614204272964850691079850791042887\ 106882199741854645705894552671509615737210695633632780080259034895163195\ 009615970396765293541218508095197651862720602248752620581961303822173939\ 765440624669358800700446819598131786936131878529482722966206687006091169\ 477452409733980253246673496381672935904345042292142150459464846868486392\ / 534910267590176087653888621764074210564274065120826920738 / 1078175044\ / 128805270054262602189327590683369084562697860739352170498395581499006169\ 149235277128993437651741787412717737201248533373750280505564225911585489\ 296720310932200731536930443811369812785854167734618610124072792121422733\ 332547765377138970638905462420926695423289043502114672273887821581644218\ 457414932485305649824460450289811977975109096562066222715368702647486215\ 207721403422123010636581533034056294446498599505872093485006116009990375\ 968192190204257994946344782140623895714408548637678410137716051755001903\ 235751558131448673009057488400916183546701085336004383170766834680256337\ 312898445882941799071372676264368626290028385117875 and its differene from c is 0.73921102238067849121992271593476473310053731106071764451184753015857269358\ 398649380288572766143302776610395653585537655072562006040420787468645149\ 117247395829443049740935285088559294455278485216720061457613137286094564\ 541412079816306842210498517422726641938574673831752219747159321924999765\ 834175682941574219036500629997518122263962994770365179716124635632556088\ 780918077540796335022496831693431326584476080474160095620870617683121432\ 183036941112673874161630114674406152868380260819578553835893374193332931\ 329174572175739484561665039104560915834424221906318772862191500844389074\ 220034563933058892792842643846449746787331489995607565880901532394443570\ 143135856420224807466801503329557771318296102097813589673347265068248732\ 816729646459116816464782392034537339796985205895124411285090149592346848\ 427970031824001991085628158989152979811834856255536949757280719480664823\ 542998691186158624015238052610375941083332015518664865568338681571104684\ 389088566829085438735852365316583164604896319027715487755062545244211374\ 876600940759966887777240912561339578108576436594870230177067952967296900\ 552845685390986125989531173824407473085434123049202866185382807685703279\ 057851752056954745464077881520629753867743175203026919049103872957810741\ 019597002578727819614168348835342296677287399626053529442949660733460699\ 120441229577567681270965675462585663143468415962557811979947793834569220\ 528332671101412371823023427347542475034257437173809406676197259804663005\ 817206218981917338534431397754630806189051783235864028589464926848547950\ 126959653350690101584199598714073100539679336877489913667349331387449623\ 677144970354158017210415486694041594813019064054149969914907275170838046\ 053541072324027559447204252618186590290695987214683275882419230659213244\ 049964012344557010590677145636494062072513588885835165609721466828028059\ 802658329407557963214870717514142377993866070953977423072654609927054084\ 882464311537916093858770757442944941485907901527443067823017569103813383\ 932631790093454063171861413160295206274246577501624598033493675008086453\ 771025432712186276612488156239987055355032686829318283357091149899950250\ 772810576100471280165085954428060086520355610745457727014217060416933653\ 316657233568851212996547233255526338831898112569661031832379382897085107\ 553674745111179884718319311082873412266417642440451525807545767789676148\ 366278867352128055312261453864487544708628502156342693896661120257990280\ 718259175967984079946749387511013931270779210563095511888988340012854850\ 410399471971686386781628677145494771603681132808771889905934759480236181\ 728880914826972853676896871658583247075287199752355961203644209943981576\ 614111597912118573165852295300663215304896061394951090209508075303674908\ 846015899735737239709004307434852275866759636322029700767918556584581040\ 225490887272906440686717174253328606731099164120419248588522867100129524\ 060173179379519158811678350454505443092776490841894170014240620153309395\ 525468024366258228718710589780055647611820186530397133950447671278851417\ 440721611555494948520685954104915069671589236812273533053841284397560485\ 216498078788676175555845631744545452584697032597798617571534147095934433\ 697327838207473491550050375008485346918872131076602938409076188636663939\ 941708780776608908583534995291904184831051058343713796370591473441557603\ 977426652975553887419169103902050935082606419158251516976644657573465336\ 963823425616024578220215150665602098861951620057506750882585294047186734\ 622274360875189880763561738076199575195634209563210444110243363462979295\ 337030861805002599313387471361410896585488738424159675631786584345571652\ 538332331295315042061760151066161790466587273148149326319882170487526897\ 510382675974621454838435654250248118472616177716198663508477889698355819\ 777976889884675618441923617179753773943559790726036941979952964790819897\ 110579398118299379279591589888162683704668022411292838103444515747744497\ 754265127326283028423800169225347269130369296341309874466403767109390738\ 734644521783504569106811540898070918891038794530553784730451461198406452\ 753434098354881488564542722816131388571065310349286127102882278088125448\ 287294567193598856509814155514918965519568829550760052044096682213865173\ 967590252723905663419926572468841106820363458570288525938096777442121594\ 122433493736410802621434371071452568657096311365095237935971353374290161\ 697374311002383562993832707096548534825905221590093619075241315602964957\ 331172900199441627343922648311364071180264906234473085261083819392738983\ 660038698590803923370627666339304473629772828090557451183762230115860421\ 333642628514977163754868452841563711970587116313482007512320633381948945\ 773439819617623594341191918561775634217627063654482684249434286200415262\ 309191875615183229854983501069375060177723296713210762172501464478739665\ 883818113258899254594042291579377979053380582477972242118459659850098620\ 529987124955695519786533886939385840370263856629160019397094536716751113\ 610058694512871222112230173090408823093105536183542298690932372433739033\ 586249922222779475558875725257504547999304389793223944297350247861476910\ 245734583533150395273640067293456762831740689873135361293621738980557075\ 217348602048538609406076356058325160813258101066709778108239149784616188\ 751193224177717932491679541433205481961198549363837289563744400204108229\ 629344028536119955876586377554780774027615625866029614686751012292933856\ 124954675859786217958134039421059100295200647813624142473396223399642382\ 242685947650514625208569383907651727735829462831638610027145999496691865\ 379705231367867968466660246824439853715118101404468209369758745958344925\ 271894802254925825809920612537274805271761760141877681206378302056257179\ 532833554690896795456848885922171381291817467650616302406852077347190589\ 447486098525399043192590399028307441538138695875932127560235191864652552\ 958352571421163399592273573623984936506368345666482203380326278516287961\ 238451391982018086092983043446524590230670664411727292739810667844102313\ 772159755155325012002990144471856021102622718812163536067200993439072556\ 223661383978350391928080402067052111580262645054397220991554618108491971\ 299670739364303917976268898706690273541937468574103488531345343052528711\ 398615811337788400366352385864129825899855022062438960089572937487316580\ 294481549384731203324495100745526307024986472474865545442263679012843629\ 828738540249523391212293096592628581728796608155618378807198490943599512\ -705 57678317575094156741345819479 10 The smallest empirical delta from, 100, to , 200, is 0.1042355790 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 210 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 7 1/2 n B1(n), is of the order , (7/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(7/2) That in floating-point is, 0.090493075 It follows that an irrationality measure for c is 1/2 2 ln(29 + 2 210 ) - ------------------------------- 1/2 2 ln(2) - ln(29 + 2 210 ) + 2 that equals, approximately 12.05056934 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 11, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 15 1/2 1/2 15 that happens to be equal to, 1/2 15 arctan(-----), alias, 7 0.97862642008071164025 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(31 + 8 15 ) --------------------, that equals, 3.8806942681404861467 1/2 ln(31 + 8 15 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 15 / 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) 2 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 2 15 (4 n - 3) (124 n - 186 n + 39) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 225 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 345 c - 675/2 and in Maple format E(n) = 15*(4*n-3)*(124*n^2-186*n+39)/(4*n-5)/(2*n-1)/n*E(n-1)-225*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 345*c-675/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 15 (4 n - 3) (124 n - 186 n + 39) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 225 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 345 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 15*(4*n-3)*(124*n^2-186*n+39)/(4*n-5)/(2*n-1)/n*B(n-1)-225*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 345 and 2 15 (4 n - 3) (124 n - 186 n + 39) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 225 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -675/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 15*(4*n-3)*(124*n^2-186*n+39)/(4*n-5)/(2*n-1)/n*A(n-1)-225*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -675/2 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 5177090423332148759387678314938919965369504812707414699163153640600038555090\ 631644682878754378777944843459461975877522289962939229231410504204130184\ 531186020982379459005225736643337186002924026741794301599395171855381040\ 490033520630791401410526208759472331178715780172778742071946511713679756\ 858849630077399386443644489203520583289006699206779752396252074603875529\ 722135277493355936724344000665387114466883565487102334594614084893845404\ 445658095331584082809914794702601159137012759075972681672752688079259949\ / 1596893634659477 / 529016008264437747550863885805168792701786187994468\ / 030986695399198411471441193101312815044559835385196502713913272158565143\ 960298585453461876217260129076154907764623797227945048161874955775744687\ 677935802469378913505936733891968576349529581837977012895420594892285865\ 527813183541021628033839802542367140040958761469643318176753658108879765\ 521593545009362774946843388952372157130692458264183326674416463790479143\ 740983416913647763611325102670873240921279492495096013208423317434724669\ 69313654177495195011363835073084936229120 and its differene from c is 0.18937710011245160452852310625773210278706056580131351565221849968996756909\ 377025510202906635208055533080109433972132560020254297376278476464669853\ 424505014714413587027903654602214549767417296139451041232046107398270578\ 720005670051737445986092991443839218545775852292612683391541923112386113\ 875443218290146111688975505976227116920084367720269750465100865666820309\ 551833441982745899666549449602780846998920978649922931060160405236323275\ 749814538115246417792301775384592198351465122573761331878769308686612076\ 830210689694477735004806219746235996605092266766624095070412878083050854\ 255911482261125923025437044693084076589097201591780139564905545732935860\ 110587780839468416573763174858758199297944505321084566323300099137889364\ 729303146162645776166192511311211638035903289486377929040010123271137401\ 564988200837672036085449381629429754865742168862069645081652550375577307\ 061093712492329880570309019638804250189518514501408426491520178584027919\ 354725560966865881567334145952400494964822601486335214249559255727475133\ 135136643569955977529085805019575839583756955509443570834916410183081426\ 129887317065140309655086234035178246635731066798198416873487273746247319\ 812684451457613153741095338540959812566259693004482373468854260015163814\ 412793402521825412993747046855867522551476820433637023850353063994064949\ 989109298112179208669448864832798063969986472653585682799336863231819815\ 761714856902463399309172143478110787063588775776244426770192033067613262\ 502684513658423952393947718176798704716752970697487457980664510199471670\ 595960719871631902518947902904181394311683386361373985944094275326588662\ 687142394169822643138664172815395220729269290447937379055351247511853682\ 617457315925272861547817309898559318006659251579333169244997505775133584\ 428034432149257420217805126100007760574103332868818608646457505795164891\ 584450100144904518536866903025292240508127705271887862574329303335954198\ 144130644250854389242717390313808177765100186375432668518156280756473399\ 896953259078229104424148157884877139855238841139059515511167236133051041\ 102276187928244815030593888826318548158381905269228065583771912961073836\ 708826453933545021146973306821171363749374563483656185560540763383819762\ 040170081664760982693689653563521177219941502562430709217804603596565613\ 787830646915914188430509169021700039369750984122850031489560565903742775\ 243571558840947703824809004784716971595612001981663612587025401543593765\ 248972214004896548211996087582735021506099078663078429924035488005209170\ 263965078297314246404773556954135662147949819901115902420082398269714963\ 197151260461490825211451246929568954376516601025471831880974465081406052\ 177942759071962122200564203529174582470191619310173965257132188151715544\ 104314434778532080469209100734127269262399510909849363246278062901570395\ 486679504832986965127412442018938231909040512076921196505265286245670072\ 147189750168579560087577784450911088533144289044157511379837250030614323\ 814221775039247950728592226270598978809712395873814050941002549014652042\ 389276793416583387603076413087285317048662683351298632481277313508712779\ 908095548406009528863565044706064570224109811300306052458666118083398008\ 054020183605630997682012344288453092235148082038695335852003203876615030\ 507859835172712078229934619526343563422330168713442880330744155727167333\ 386566138828063854364083013776469137872922971923110236969085424123662663\ 955730227166358312802032552815425556665750993523542890151176598425053915\ 200455934040119559263729772409956469965184459222075331092704749462052808\ 967706373310970592142223055895488027074028387737972156012340147577987112\ 370018737827626709099130043123095566313293335130934291631955195451040514\ 555963884639400116127015791696617517673265414324572133264628076789128951\ 612496923894976626048820790670864192692201729434085146027018437963927925\ 256079892219450792549757446205421975295538327533053832336736149923144620\ 885318093556269168912206074720635698818096327841434003430935985651630384\ 737343008585121870396363217271055345668528751319529086136551377423029001\ 578765190267562101967197614809345953741428114972026628155348889024277998\ 163177780424422908727228506606253187363704587853946948009557363028175817\ 091920721228926099098872820627193930920646202399789691709319227660639964\ 128219087259520436771281711036237086162529166425805431622829416203176449\ 861213420150440062704328109158228204906277655124874491382808657414166553\ 057045187284189871163494742540170739362965015840386360573072772238272557\ 804181077876755237866704510832979534662046408672548074256174167822606691\ 209084420585579693890092666017571988367934494930119008299654357088802405\ 110462738249187736290454166808824469700321649870370749215446996620724489\ 702005384483419751662210278215947684624989426561774920320926026629438189\ 850040984571501169623492678095626876215951613382705389171600285362203128\ 258204974734808196465905525610487466050361395238910402185868754284385507\ 118222950877787468816625224358474359133291268220975742345409712586923379\ 925355227204804776049432534461910966478378959683754347772222953752383474\ 029410753951195414712287805089117989697450807474093107966680235511475961\ 644308714825688125014794093210163328572007938727491873359696590282233287\ 890734718478709921489708656590909505300126016202996196165631551209507386\ 439826009054942706093118765182570554206503888161671794446569419694084731\ 741318610638902556980203731237816631321535979853995765508180178005275579\ 686149650689066029383494791812351508994430687248393868088219990589081358\ 348367409599127315201054795533679315772076529252664781253515036488217084\ 359885037148253494777198080297142856051098384380155242892184484428377736\ 053560841285268954347717107228386493755561655900027194648790784604978938\ 480351066890571562524275152767765173114471409105271500355590740591168062\ 405433895335799029092889487048466768354915559010990804882776065782449583\ 584872439827888922657933062401532377344335990563969897037510597021597830\ 943727691880643516197440615276157825475087988322542230844850520671817583\ 936188375463610909739489635120914532334992423118548751402520746262796504\ 977275921685072143336490542579837084476394482769878828834047387293392077\ 036112376836867316378928313006845716611068498563592592319149248384728781\ 340308756658583462894951818485533036489733290710860643337769349851207507\ 715765790910334584912936652228275932060540740260508442947494829812251866\ -716 779962062237905295 10 The smallest empirical delta from, 100, to , 200, is 0.3573749581 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -1800 + 465 15 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |15 (10 + 4 6 )| |-------------------| | 1/2 | \ -1800 + 465 15 / B(n) d(2 n) But , B1(n) = -----------, hence n 15 1/2 n B1(n), is of the order , (15 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(15) That in floating-point is, 0.347138537 It follows that an irrationality measure for c is 1/2 2 ln(31 + 8 15 ) -------------------- 1/2 ln(31 + 8 15 ) - 2 that equals, approximately 3.880694286 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 12, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 16 that happens to be equal to, 4 arctan(1/4), alias, 0.97991465250745661668 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(33 + 8 17 ) - ------------------------------, that equals, 10.432735096002347748 1/2 2 ln(2) - ln(33 + 8 17 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 16 / 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) 2 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 2 8 (4 n - 3) (264 n - 396 n + 83) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 256 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 392 c - 384 and in Maple format E(n) = 8*(4*n-3)*(264*n^2-396*n+83)/(4*n-5)/(2*n-1)/n*E(n-1)-256*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 392*c-384 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 8 (4 n - 3) (264 n - 396 n + 83) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 256 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 392 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 8*(4*n-3)*(264*n^2-396*n+83)/(4*n-5)/(2*n-1)/n*B(n-1)-256*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 392 and 2 8 (4 n - 3) (264 n - 396 n + 83) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 256 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -384 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 8*(4*n-3)*(264*n^2-396*n+83)/(4*n-5)/(2*n-1)/n*A(n-1)-256*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -384 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 1597448602488174686294654493715929341569935598680778352360924279515448295631\ 593330921269869055827004703827609952812590059764279890568535734108867738\ 463790831450503165939707147321312881462739358846356659699583096354358127\ 082469661666630397058926309232961856634916018992008108077109539632000734\ 660322659209103620130020693995938256639750401991994158863987368706546589\ 690016187921030912212774544990818792879076329819517095560663755798106656\ 234571843449378889168101734474802855107270134536941561574708779993903138\ 008197783004034161629230302627148490292461767880395161645363982959015562\ 269164131827062093464382980698660917217610859407636060497162204135988683\ / 2 / 163019156658239577926313485502367439578756130267810732743302057328\ / 387762009166957043346109460278926778904338813738131549047496072795817091\ 290119750600633930436850892862031738088015590838434583901800231785138938\ 010728561177077220781096624026917296896545582764628856245704690656548560\ 519827884120467629083040260123385026007579749503029910217457525117898408\ 403877132250273693225029352303307740381033509532956588186803182956847928\ 384280622180478746557409415993593304905316466411895121023437730556820855\ 881937154037442552697148440591071186632169252754784593833149686973164407\ 365322257935574611395525532870695694691702803078018283922811546941029964\ 27666564625 and its differene from c is 0.25795526919275036897340797467528230640496415480901470006915197251457414106\ 680913581203107846812830020939199203040238714934205256094115545460630811\ 881990316423269345422382965286182660006639368048119091199719071929738732\ 139702905471445947283327752060945545286504970970414440169470367902563632\ 531625149987539532287294288687168897749608692836151320734116161422093125\ 105460424135089109178916328782429668081976058422918730011120649682880325\ 582019150351870906213276412338835964242408252139694795810345625006367598\ 950862353642247741086490535363981112188382493365859523382395919787050668\ 861420916484741444363270567694334578460410979138187126028977856382696977\ 870131333650103278035160856442056825168006798297214866326527252344857527\ 038052291557118845650499195864858522316123490768382736507879361143864582\ 898630513102780560455363454607110227975575462518498601315111421352826998\ 025203933739942023133106687764338147276639615875811966889209613892054342\ 540349627621981197802173074060231265920313790941035503895918623183527535\ 781320153194443858504748769106563556316867905995410596119082806194913777\ 507082274481271540639865706006240372343461856631372494378204768208434147\ 614208456468318979623078169782546045875458789375825927596165643058558535\ 886023691725998784051672180040109575927991929809827971799878870980665244\ 507135201296970691959999091892966181105442766193658079051868180516192408\ 682322875818833032919604220346478416169387373436787129753913458066169732\ 387722204051936568725017696467711622443642973298500497497034647361920312\ 258385150701221458292074603175099439206314786347798406598916751881864974\ 229407768356739968735814224612143356606253069678608102612016474615391958\ 906218587558100438052256307566405822534603734424227602635376312736991108\ 781925358870055691992494590626334552418706610663047189383097785108575535\ 730211892819245053218596005596253757738674610600744367651958473134070896\ 479978584562718682012124035896087953505143767694300397430726282294912948\ 964726618189440389182378121271780837552167856289883441766911064312282018\ 361634639498593483948292211754570936022916784325772966303330242239591329\ 565104614449112546676778688671747491025752692339412597142684504721711517\ 253391034084591391966973638744771154226944538024631524024643309685848891\ 486819996488516325047609755254270319799803885787727318226187792581512000\ 350507396529792704254120742596847597202126146779053756612689415023628365\ 842378890178821300867823144304152098965636397118462074223498171346758546\ 131329919766999786492911940722799678581120444090362261080720764383987539\ 742853065016656279794276254181020161716585318108442062739552674273115410\ 084452772287590760501208608999504250537185015546110750178922213684452899\ 315968783026165810786199583718386080307676199594895771867862782680210874\ 118327429666885924225087496308042925600062933591652421848378303473524052\ 330201659107705996016119571199220008388136451816973796552789922203584086\ 293864144168880517838423505884104340508599910289605915943094519619745155\ 379194366771405589129355603185298063064982828977930300641783559049216964\ 229644059348061138650311023831327928046276450179598108796303325003540098\ 878110129013780597767151150980174451384299450956100229338238189895876364\ 135395969488087057133050325763645465994570616237153629715003195957803062\ 491245729112250885930561211539827165156525443275601567743344585672735617\ 242958052674108055702967893874165771356545999963487717471398010555982699\ 732743646591004862768880087810636941119508072223385646630268213698626522\ 169281671704169741497461438050768239647242222306581321307824497104834290\ 433112941860459565863820810347983555557694550196763301261134690946227034\ 571548758518264501091333757416558676077457630163350705813201620387142230\ 653747644846594251599014526728571470579381552719995960964532988800719431\ 281815503250920341259922170419150775637911701156441468379303419769958095\ 951884112637400075317166443563215636404880500972736940854933049483315404\ 796577334522159936299508818212091670277470247079441773617019027219662236\ 841429582043824038592093831737095289284593931474511579914506602286943923\ 092527608789906866619963693484214824480204546512429948811801858160652994\ 236394165873755646684117448770449269148921421770052934996451275635188315\ 145654520966672402967068602380463001968625169679554198171570227968120452\ 021856361894028274060627979363841511919036322751070324595099556145248766\ 148264682616002518221085177328400462654142496940961519224252245872317957\ 849036015073836261530225756232759724446126676665831546775619179457709635\ 200916140157549953804903994851129409655455024443650140142147801452886818\ 887982881234105266712613609385245777819292039515793507612567557413449227\ 249799726260833802395991938000778419380804035870515074963460138473022204\ 504574681179702846559163335420697213259163666803375347448329235482282708\ 062803866050490559283820553896013406595022347661507055014290003233849487\ 450715890381845324009381102269081638758046528678839642126290125082814136\ 508984506342680867998457555499151968876257732723363714090014134935467470\ 880903895086006143307170871377376663496110524962814288222897401767121669\ 416700762214207160911888697765707318799542373779596095731189576199989831\ 379040464306076920540111697759179133316709722212607998565793094686920307\ 488375193577904779460252234749805719144640984538199519606667877471566246\ 094574571189447693748830641971789682378410342113603195961579026251252598\ 773046302381565042131756542328072497728944315774096613740663752683790407\ 039032838181987698639066760859410763153586367410565145605697278167773763\ 182691505044380913806962037192501050471665973827835704876108426263181515\ 871901882263329973193266780565537069889597415461008970270359316202887159\ 672430416147162506633687175537882832890877395557968394827856746283164783\ 211301590781441689652695278187445586668006784836860576719509855684530015\ 197600801155277065591771144097574544749080872175707829672342529028354516\ 139904186549940099929787201293186181769834396300768880118510010066712328\ 400323855168816813155065084197109896407042937936646715773670497769517193\ 982987371402337701796140172660679506089140600317919234067008355835347442\ 637370562137952950992913049085239975119208540708326090301584424117399136\ 262499350930498797964836311655933960644221732156461885450068774490656978\ 676842842144573873526364950563080941416673179144900013772496107595775233\ -727 2628300 10 The smallest empirical delta from, 100, to , 200, is 0.1151896085 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |-------------------| | 1/2 | \ -2176 + 528 17 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |17 (10 + 4 6 )| |-------------------| | 1/2 | \ -2176 + 528 17 / B(n) d(2 n) But , B1(n) = -----------, hence n 4 1/2 n B1(n), is of the order , (4 exp(2) (10 + 4 6 )) 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 ) + 2 - 2 ln(2) That in floating-point is, 0.106013701 It follows that an irrationality measure for c is 1/2 2 ln(33 + 8 17 ) - ------------------------------ 1/2 2 ln(2) - ln(33 + 8 17 ) + 2 that equals, approximately 10.43274304 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 13, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 17 1/2 1/2 17 that happens to be equal to, 1/2 17 arctan(-----), alias, 8 0.98105639035322571815 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(35 + 6 34 ) - -----------------------------, that equals, 5.4635343398203729553 1/2 -ln(35 + 6 34 ) + 2 + ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 17 / 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) 2 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 2 34 (4 n - 3) (70 n - 105 n + 22) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 289 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 442 c - 867/2 and in Maple format E(n) = 34*(4*n-3)*(70*n^2-105*n+22)/(4*n-5)/(2*n-1)/n*E(n-1)-289*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 442*c-867/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 34 (4 n - 3) (70 n - 105 n + 22) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 289 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 442 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 34*(4*n-3)*(70*n^2-105*n+22)/(4*n-5)/(2*n-1)/n*B(n-1)-289*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 442 and 2 34 (4 n - 3) (70 n - 105 n + 22) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 289 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -867/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 34*(4*n-3)*(70*n^2-105*n+22)/(4*n-5)/(2*n-1)/n*A(n-1)-289*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -867/2 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 2814137063231986166269570251203035463609078954831868634361718268360147228204\ 307482734267633319392742587945082921250115875411250022146618382383133268\ 973914096672911984100370145862241483826600279985522725673045095513335678\ 917993804499700201905094363451644290579017162895162936005905496566020611\ 252187892618112624152628255648836738870923333098136507290553481960191488\ 367730863458788352230701135046210124428192099490871731796226377304414616\ 670099770321190306393005812810878339575637642640395750370531444373597220\ 241221555189805151447645366283978419223623600314649343732584441984889739\ / 6929037124 / 286847635967058585095283686897124054015056297588597292554\ / 803052892075168434785142111896194155060095333332571210701203610768415839\ 493262667180733719892177915550248799833455182057849518151687894172424232\ 323992798590351657240642727546331900660488319173167890140492238680045221\ 743677804511529235029402102760841403483919567264301061094692169953280752\ 685767590791046626838688938526616880885309418975997389207729226525160006\ 007200344634745176980997400286521761871110336538269429245962049162399518\ 435657520575171077861732290838963870396529256250118206426048602519877452\ 22165099688238393797774091375 and its differene from c is 0.15340999242726235136186291441898505474886198196973962801089980045466399093\ 857986366090674726089458104927511434863166603546393383799278895060493477\ 670103208971041847080002601519346490773530447118884062470290814069843335\ 105214919028425465774514202452149944339754870590501856800664667294528589\ 683890918885547502902801337329729150284686516873591121368399766107581884\ 915971300567117107542332418499551599706901528392344153754552139999786048\ 149378217317215525246176388189350432627107929129197464333004221399458899\ 280151481914863673917091539086326299330470766641121197552348823291905451\ 374678993152998123952207884797188318277132818437003781658760059880951359\ 226031815717288570892985735953406131930200103043422541259030613837847481\ 562147336119985489726201805308180991016029274084197589638341663245948901\ 086260533173830175531593937963492278121571726093520205077161988334057357\ 359363204444740229093603366624161875486824750083678477764449246165242794\ 488401547481846387837221492900387988199907106269703591783304795472257365\ 127152011982901032705529388721173790717661236827526801450317756610075919\ 112255723510272994810063351138363403300610923036379037365898670766886606\ 075251692600038309563280060290820670936626001946422943918165186065057990\ 892914834134927327786074206499304954813025663320044866659956851132486243\ 583531549773515544006647691828502896869705729390285047783336421200125184\ 370442532128741679034207486202275447363917288489430371867234015168732845\ 907310644607125413825998492370350884313822219125358226932846748395582117\ 296980324483962571277946635900761545666344679635181992274727847583044469\ 091098924526718141421715501102285305612100527875737589848513236259195618\ 056686510922287341670827826753376096478956956261289083871921671214261451\ 011755858973344827241346087230603445704830220128835120888829894678297989\ 740118895244865050239713352332146684110106379695655890839572625451003083\ 783508108079602034011175446850468602788871804314739195920799547487328977\ 401736893494507639497234145622886101694087978255381362949731291070556234\ 375889394337071474881809907490580444888276427284102939285531095585636605\ 366066536228290533562984831906278589390282138764724991069418535527508595\ 113318985648004767109986892555853566899187992286015051664928940394297575\ 042407729814638189024453582309642001671965327784893877506718977925947000\ 147361002828854401180683041986834607867994081857835368488320696084465469\ 276814906698785759844227663364175830658127818417757226736360528986913598\ 600845602914954639209605250751072950835394055438580981917191103879506292\ 293669035471500179166459971433376685243804684802027222980961597745663411\ 637128933624227961270103307452027459260589666456634320677808946202308660\ 327258488292139525754643338018388038229148350061420717796421982216151948\ 420439475110015856654233021967232287473312593090436921984057560571326775\ 409133971387164772812016531554669408467982897786692470213444630762821341\ 662699330154467602390367267588676706863744337253894298465632582926130162\ 011889096793305268518170819580409489412353198525412634413554711436348104\ 577435269615610336931029885678584903464811966881314975460339779271074995\ 405222244638447767192153235379283894438642836627403132308026975212356322\ 447044381402589194285741052076379427405178443566831135038298285596557974\ 835516926614335039225956206294699574148659647591163587991089638376455445\ 762384448083898539346939471644947389851349745701323215167600300240368698\ 242789731540677292916150912030893101847175064870982716875601382925387758\ 357138153791289886390060799164363512673786953141467212093951430155444282\ 648579995017872541752214611195687107109399171349519958685796425406545221\ 629935490202141250482639337759159864311951503148872459025231912828702290\ 669055760184681245569939830708531734418039618790117799671402690428488980\ 502638931133138333944147869066625954605622086857145513581223936925448124\ 355078911861975662794820433352051335323177259210558121476020594775890461\ 245143909615348146955115771414161694968108731295318781787983483627596639\ 575520938529640442770507668516554535136198348994774575621205507732710428\ 382177621846231642671693601202197448360846258561602462895951970850597207\ 317373159002409139223505543966830380560875873254446755178195128227589737\ 600823730699689626291346427924109811114245446936798144714262214032453272\ 113841561073965696947616985592389414118631079020229601165001658026948922\ 716051683394360097199323517874907602459947066100693815800456360569555891\ 486429642188833182247256239442058728670927111439338052972482709846881186\ 255135675799540986275546838977264211926217208056419381695406636678283037\ 553585299831712886267406857700791252995483425894340265407549334924921429\ 818508604340680355709280576816013926399035829526933135520554849736090560\ 396866841908188342333812818537814512378631982857376115135468907851078192\ 983834138385683587556758663561430198255943805511350340481233872711848225\ 730890302448578517438371171677883631735078299949517437459298428554662021\ 729056399106975640507481206569422397505706940165673091962103049923292598\ 759214310477886229568303410134065034215080098226408433144975164641558150\ 795853465517118390043647881048876353162998157167308208833653924888497618\ 775821120537454037775647146970995846480624029181039933802253918653879603\ 923612942724080910183201488347190281608583216300437421436629399783208891\ 601751315658305667609190826008366978806092032821591133539786436225660298\ 823772075191757926726510310653728627670554950462478028478899342748709063\ 652456976194676672574769766222542793981839705625483407837659707346359285\ 083803349214959274535260821448853290718849274996665875360940020224071312\ 909128124904793779800794367592043483375348181975583490197776832404800999\ 121921110239509586662948126053514138582818350422302090897008765667105722\ 479111418199747025361653715159390197966418709573261891102561737028283514\ 960600513118534923425605333339132849958470323036784446527469446784127208\ 753068551525238343025940072931993389322571011251046538944876260526661946\ 199487971153220780115490851062101867781382966917583216909137659310750564\ 409051146730958804190105000476178458137502309861477560354233912307104666\ 985132308283666103845995152398147213648765853450074768841485097645238667\ 441030021466516427542833431696390493735785075451877983613146097910798690\ 612926115654210296380656771473705047921379668913020392463193213423754 -737 10 The smallest empirical delta from, 100, to , 200, is 0.2364328707 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -3468 + 595 34 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |34 (10 + 4 6 )| |-------------------| | 1/2 | \ -3468 + 595 34 / trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 17 1/2 n B1(n), is of the order , (17/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(17/2) That in floating-point is, 0.224037650 It follows that an irrationality measure for c is 1/2 2 ln(35 + 6 34 ) - ----------------------------- 1/2 -ln(35 + 6 34 ) + 2 + ln(2) that equals, approximately 5.463535482 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 14, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 18 1/2 1/2 2 that happens to be equal to, 3 2 arctan(----), alias, 6 0.98207528252699838771 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(37 + 6 38 ) - ------------------------------, that equals, 9.3808596317363405429 1/2 2 ln(2) - ln(37 + 6 38 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 18 / 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) 2 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 2 9 (4 n - 3) (296 n - 444 n + 93) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 324 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 495 c - 486 and in Maple format E(n) = 9*(4*n-3)*(296*n^2-444*n+93)/(4*n-5)/(2*n-1)/n*E(n-1)-324*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 495*c-486 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 9 (4 n - 3) (296 n - 444 n + 93) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 324 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 495 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 9*(4*n-3)*(296*n^2-444*n+93)/(4*n-5)/(2*n-1)/n*B(n-1)-324*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 495 and 2 9 (4 n - 3) (296 n - 444 n + 93) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 324 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -486 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 9*(4*n-3)*(296*n^2-444*n+93)/(4*n-5)/(2*n-1)/n*A(n-1)-324*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -486 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 2619063903667314899178290382217796390121148359174755239811763716458904769839\ 636053439806006435099009551657503512444066375408022867261570006695488267\ 378965173318931564558699154888977424643025780310199795840930958936433134\ 529572073441408263293876698086078654828275363442553949936685880077617465\ 595529833727810130700601476048788806914436667041786944749524016339253098\ 755030690099676319418808897266468885918900574228231575251273843759399002\ 318440943734921292729854170045888014219919863249585338604285758922611660\ 241583333685946830179011298080878255210283230271515923635281139175392779\ 360520157628321083721199678404188332280750756503280531782483976134318169\ / 65702 / 26668667364564424796020300458908289083390768737255069592965398\ / 716462213729695781086905354135160224728848932821857872078043892477744428\ 302519082311144719706488895800761225945801178958143985609468950198036031\ 728137972154207990993353242838092109591383654311105405265837144204310405\ 397296724746703610034753130181625688300251229205108212211821368382627825\ 444541731541374516786021297376490240402079480886895801369404488263831890\ 668191807859252382792839466434625567403325211919376240141255629986360948\ 645650552166618901518146180707614779456423915707924208085872053154334510\ 689121026098342788703134890950778069257987187091667681783802567867812459\ 5984371946027247875 and its differene from c is 0.33805732445157768642118529688233878350686818534640274508922467000820128083\ 466236258533012640994576718215152179637768554358078203327184605236390685\ 824635749014298613220391051894642321618165939979346877314743651282233540\ 635236169389426164073906170295068152310148729335871001920338038271284688\ 043579319178348967624042613550282278471200876650323053044200952968140373\ 048293996931713407928052395272170567829282247270861677874855925045024836\ 626736587981539377696995429237551630293968142862958504236939848135293339\ 099948584205020533861424447153569808667269948454663207049939137768296459\ 717822504723144934610226045450114556594750786934766435153526836391154400\ 442592034272362364968187817147858619601813729653555613461441993417026546\ 045039071079836915075194083450896181890579618716636468751403415787976271\ 170422478061451844513169165385191849589464026530980937629399812833745839\ 055951096495355130595846799018959656500396487390914053573547888405230470\ 798557707158259074326168770584042670040141577782260050453065783523082845\ 835742220637945946049915615798646390046035761938316750526022312900941333\ 632451553335036553525215044105512442039027966568494937402489148569848441\ 393611665029666406368105209215836121741833301995065685436014426899811257\ 231615953359202937053341268915498408364745589119050605567963113577538156\ 107063324890983255169607740491077705712332435077157097158247014731516724\ 934123800756388984272383541389966260986470046163517461602956285863186530\ 275141553616568611799713002441337666466603876734496427417465647293434477\ 173885891867618536692445536749725840974176814080768274854786283462205464\ 059261995573842274749693707041720433510310522896047170290943439033456726\ 985787316358087230931720351812794820564196761338464188286462438605257674\ 215067595481999242647616949182341825722622945282461312610369899369178420\ 242578514561472791779129221527714662413332120929544324219334229061661408\ 870110329147498990425853261236027616988648672855568807457751145110241876\ 134081483805773724427122465520462220552433313905473257599551773036088412\ 364096106993722465750529845472496696802462530943155058974649596832181455\ 260141131302520830780940875567013605941790410583631956049784412392653529\ 660850725309069564291396953770546342987583066994865553621145947670439239\ 241823396665448265496125397372772133758143581090446816072159012562589672\ 276210297740312615167779904555599349443296309081596025790411591632613511\ 138175438750203048841805650747412722427224052861434186844514163058214319\ 172527534889071773449509937777683895572638519228621910810715833396121236\ 721292480525167050060611162145170639678913872487063456896162231503462633\ 312514682266822549783116211439095758568456147602735379255666713775929613\ 829708913639309041014540612754946456372367751000949385491769707247391237\ 897974385761979475041865785079108829039604779519570016645813604077276729\ 587152836049562863866494065788560021506352611750500574992805480864313089\ 826834335426314059630685914606020164582799493320392873373891898122557841\ 779337060974096856689229268551899492101628346794788346705779796290723750\ 288311331692240711816435727631373547077044591889044547005378572522247920\ 715254539457024107732149436859765245144261112567059075556662433074017971\ 875998179830937223719836581342808810087173603063599841942845133002812957\ 314171144689853683557953068291409395325129855128748579919776705256080508\ 400680211251001318973512098841848752398753852336618523876331597579221821\ 667002707143941046212939352542717150049999994930237773022183153880894407\ 742672930210234282887371736510548200440849980702021238677736515555226680\ 741968141271687393228834908888263837421791780657771916359308795196542341\ 403736660292329522258184683349398111524604399019927240506316881558573730\ 106408349334811158529726908459389560720653848387961611849142479057986125\ 924169122946447654224127578684887866079685632930088872584927142212875571\ 810549188344252101155644695003129904557185277648579218843969488921892986\ 100345578024348848033449715791006724006206352714835524744815838095134155\ 534912758533729252548262894718399387551421282556022974534175594381707619\ 193090114832437909414188825676265177043840379655885143588484229629534175\ 529957669685666905876458333149871284294951310424924305799779590873769225\ 557882073404623945135203036648670415257302953568786739192239791639510562\ 889936234581742573696022579830706566278664931847795230955506458361433646\ 637235586649085483079294378450457901684699006260041990340232730470771774\ 565361816386256933997850218714955584955151235530857895156263361301358845\ 767425732806371891328095650189480287786711410156510843026391394375847985\ 365467900250960885386986947394835135045563581531451803524407747475003826\ 526722477878408152880291583539169433307841597580037190646446431327336241\ 791215412192310684994667635889717355773909384740437104956168520038568939\ 375490802767914840634377464643254914863623456704971685470815856149305900\ 027098560115299764846390935113104547483973411664164136824624661808899973\ 543693088385333829931500705039798670352130449900846627083779324624299011\ 701673395650417446748869707709538128542171085638294653526528008528842503\ 801822985330268993001083804215252050864313976080553688220251351196125623\ 301926592550694072172873795037660150710195676704983643444935476165676419\ 249016191589241843388560610452696404789827742691624527599468875911322125\ 145955473786326745190335785480637553306200432773521499148207345191915158\ 369668667483799018630234352531188773424744759526291240948890173087577644\ 762195564367172136344002222478278218737932970659233683906729808172712665\ 767634843315235766238620770111578598294311691100294291405514164708494825\ 146444929497195072413356220413444081081848909840258523873712310758374426\ 430383655533552203424947925441900033316952769528246157180760431513581197\ 386500293675716456463006894563250891624458763074700344886804786076205316\ 915068078425737619570251948736694735059178066139252254649462940487207278\ 870732163842431152076717968569352201458600645836209336261740556392706165\ 176445226888732633783366067327596648519404002343472288492114376632617763\ 487978413378530412047571995165291784129435427298432460288488674748134613\ 765121389050448093490612384406764177915774734850402500306076265305948844\ 168048336286626450618355088464460459303947623156009417701804275214286661\ -747 07984612958954220181544105941163368402462034485806136646042 10 The smallest empirical delta from, 100, to , 200, is 0.1333219346 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 )| |-------------------| | 1/2 | \ -4104 + 666 38 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |38 (10 + 4 6 )| |-------------------| | 1/2 | \ -4104 + 666 38 / trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 9 1/2 n B1(n), is of the order , (9/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(9/2) That in floating-point is, 0.119319500 It follows that an irrationality measure for c is 1/2 2 ln(37 + 6 38 ) - ------------------------------ 1/2 2 ln(2) - ln(37 + 6 38 ) + 2 that equals, approximately 9.380859792 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 15, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 19 1/2 1/2 19 that happens to be equal to, 1/2 19 arctan(-----), alias, 9 0.98299014672362325850 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(39 + 4 95 ) --------------------, that equals, 3.6974006367249319106 1/2 ln(39 + 4 95 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 19 / 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) 2 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 2 19 (4 n - 3) (156 n - 234 n + 49) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 361 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 551 c - 1083/2 and in Maple format E(n) = 19*(4*n-3)*(156*n^2-234*n+49)/(4*n-5)/(2*n-1)/n*E(n-1)-361*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 551*c-1083/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 19 (4 n - 3) (156 n - 234 n + 49) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 361 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 551 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 19*(4*n-3)*(156*n^2-234*n+49)/(4*n-5)/(2*n-1)/n*B(n-1)-361*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 551 and 2 19 (4 n - 3) (156 n - 234 n + 49) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 361 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1083/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 19*(4*n-3)*(156*n^2-234*n+49)/(4*n-5)/(2*n-1)/n*A(n-1)-361*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1083/2 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 1689362561621391473369537526243338075173470047058127299379457825404431240721\ 157099564967731700614891410443461176100088325961909778311382058610692869\ 618075517651999387923799334311859368331316947535256416881465764415863151\ 304079895550581836579457269388300823098480024405748821861467492785951057\ 746177695410676247508008740915308657910703171723433126107800617732880471\ 938810800545239931807958239345579776036837443191824554944046814192355348\ 556500779546859349680401349309213441624210613020191990703798987764584443\ / 9210321780493589972038665806812464819597 / 171859562097561017669240697\ / 103040691223063057221857156527181337700181482098502543546006449695729765\ 117381591973015449019763278302407841036454048574165306122107387023657852\ 393993672826520808522641540014418811235547368371833492280182615773135930\ 523272324668317440710162549439724903811327487643394447437753784117652879\ 608257653096001511787172769592873048129547678169076253935692443536711870\ 473363297811318072674860253872222497848541918607333489099529673782426216\ 648729459687472965936578847842658507431647056341121981463335769119933252\ 26989073425184000 and its differene from c is 0.24041968725306049801187191568592458239788053770944027685858714096048475257\ 249453455459731209022279928465100166649368344056374118300187964998030587\ 874113437199405486447812527362640412558494616706292515392599958546017314\ 298099149558293018276832109949779821095821563241131361099663264577187521\ 754122503733585913783920431205907920636036628755596261065070177072560091\ 382319305764596621308996694960460050954101445496350665330258688543078025\ 718976578359964113464689414191634968707582625284302057131181994202516325\ 200469026831103515312867614223372757639641942124938210514234529851641285\ 133796424820383469900768013024434781026171943924948383079473913818030437\ 023406577634816927647539471309665922371044094883465194682079248323912861\ 034497846269087754622715723241628793862599879328106529446515371622477790\ 611777994007066141983494497460390194293249913565351187148550061224472621\ 464224715601961174437243866337158083516578520255746249308560224216616960\ 163262855747643555386279267456076721535716302959800516642503824296005697\ 917557534706694074039336806595625256762833918492889701071944316965796281\ 962946634536104772204303011425772315360765469477730469464676404900755009\ 420033878938629928332328347002918463968151134871141623302384770188640136\ 339315161571461668518174960851020400050002039904288283343870873883187042\ 331530610301704913130122712259589144299610339041146117062279936293083505\ 802128684581925275803841419532439508643179082078999957783206804237180123\ 842500458922574913528185295017630651119121964756271373118332898085172558\ 445705649672786833186324587706585316583127351636490457583453987003848848\ 027513398301889753830778210659777111433475578986112724002980194477928598\ 747431077029319669268846655167170164216648493346531481305877822066464178\ 651825476425287979093101585155313859730190655326874296466120825288879468\ 235726059022131433908357905597538857414574254618085518740636487862788358\ 364956506684051368351972064495677080083034256789492341654916893177901466\ 750631435913057014247303537908429344950320035425041701922489010206131527\ 492470536812711392605024937335393414292109950399075047347239528064061131\ 394457909742431580166230172208501761818912687135626112222611911018985509\ 454508714879742754543029565968449607088425332357603518129965317369768026\ 442234824737131300498571730787797234172931438099490784069604567965588736\ 076267453640397702191633236182981485364029796978720350747825703069970484\ 067790051308907356578495163275615252416900600640697815550192191792735221\ 285984758973254808148662405367953941670809374585080175334675190741261622\ 212167675067980950599251719595675612942869010240803238974328241340342860\ 317579415783972737813661637886043311826604902625540003471193915767126965\ 954214959964494501077181409193448710652332435555705392290866859848513069\ 474785639962257863954665314421608046608365698121616606100971526325924049\ 669195782241852541061214210245841217907422527919473657798410990201870009\ 776291284918712543816534527080573547899872474377563573772906006808480184\ 557968574282747472708958761625373892683972638266763659071187910700682533\ 500296407771712218580277145216689179302162456031911840213050869131066006\ 371578452517261567467395996130516995363807751496140732181803536729198477\ 724250423257426948568184230057545178954178756195810843978118366697593087\ 177323370042151136036601884470022073677513888037701590555838388173189179\ 562430261993582421342086523748635276559002320186143026367453543303337048\ 882206579503221593862355093474606158559377289783453743449465597277703464\ 407226817747481161996292013225085533650697572005570956471802572852563582\ 646398946935740982480218962047918920725539255415151856202809328562868136\ 505843723528575470859755500890479967599700198501188720521757932385491976\ 735775905801987335613550778656452965465866496844902110689313610042134488\ 963216465304708097983122094317751742572901614478194148091861343258652417\ 486240883000650630130079529459840622871432012223514044103679462924609898\ 801713927222697175826894473729811534570174406891240444490759935855534350\ 745670134641278626965735629123964071532690913801279993145363669937687605\ 297982038057583818241149230981411727769839769937000560395232479390498856\ 815475583805076319276352702577241587938300712607402137475278899981357783\ 313543013911135632684938081370143205526135129673650245941524606940740474\ 941204285613956265584907256164721327015135094622264447957012668983636545\ 336196611141498715107402287562882146931444963015388368474907116518081222\ 553258783177446726681965472403566971944098362070258479516903512053747774\ 957846617865893620780475804553267565368278463288838009348913163891798808\ 882301874154761290504613762780492093883771390127506771281089680901766503\ 771877593173017618462054470946780138133243281671758201498755966956869589\ 517014436913083548291868474355241060846404550359799648498509400681382938\ 696351922116873863653328354266649590216527327515791655629977312877127204\ 836097972555445990267662807075589067078465405208964535030324410587924390\ 972725510401772811946254369511753180916414601222271697730154381179686310\ 841928414162762569368115888396521489385559469844454075709550641978888472\ 516980143609875024392026116740839707983261083257489945971752306327376060\ 047039336864486458302030089665299163082125182394615878944287373934308685\ 346908891129024441770185246811450213047789172421844158993336187094898491\ 756945021687364678354744833184440603214065905512965521696690425390222041\ 861077378263593569062907968846428175317602103952592672566897852241170157\ 415321081583600594598590129564633255860703296484237335545531747507290887\ 805921333702676568663684686035669141096397163677228797798225482758494062\ 153591860416959712499599206195655773294006854488260683291499323271547721\ 603817576464619351871071042573641920458534515386307336539106483427585334\ 345252292523100007727621332782195150447959789640481300010634832825254223\ 868984000255668383137741148303691416241316937666519767106700574465324182\ 483325034873045313162515807514529071773104749111782290238352332115939375\ 922955877338974148143592270201625383584029104881803293505787282955092384\ 367793380459557752503621822409050459223981939585997826244343373232367294\ 483975039278188873975359677710363125059177427964790516734993139019498065\ 434949612579673169038026034694221881301827713727083407635571742151246726\ -756 39015350246024195954929754201689126633273084067683 10 The smallest empirical delta from, 100, to , 200, is 0.3777047007 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2 | \ -7220 + 741 95 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |95 (10 + 4 6 )| |-------------------| | 1/2 | \ -7220 + 741 95 / B(n) d(2 n) But , B1(n) = -----------, hence n 19 1/2 n B1(n), is of the order , (19 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(19) That in floating-point is, 0.370727247 It follows that an irrationality measure for c is 1/2 2 ln(39 + 4 95 ) -------------------- 1/2 ln(39 + 4 95 ) - 2 that equals, approximately 3.697400874 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 16, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 20 1/2 1/2 5 that happens to be equal to, 2 5 arctan(----), alias, 10 0.98381614337786891484 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(41 + 4 105 ) ---------------------------------------, that equals, 8.6379958426045497979 1/2 -ln(20) + ln(41 + 4 105 ) - 2 + ln(5) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 20 / 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) 2 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 2 10 (4 n - 3) (328 n - 492 n + 103) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 400 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 610 c - 600 and in Maple format E(n) = 10*(4*n-3)*(328*n^2-492*n+103)/(4*n-5)/(2*n-1)/n*E(n-1)-400*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 610*c-600 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 10 (4 n - 3) (328 n - 492 n + 103) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 400 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 610 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 10*(4*n-3)*(328*n^2-492*n+103)/(4*n-5)/(2*n-1)/n*B(n-1)-400*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 610 and 2 10 (4 n - 3) (328 n - 492 n + 103) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 400 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -600 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 10*(4*n-3)*(328*n^2-492*n+103)/(4*n-5)/(2*n-1)/n*A(n-1)-400*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -600 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 2169792690195568584167250928165262371665046711504816232687374149805242615877\ 240140255363394361997686830747967405332337553129561673562000869254341761\ 279680457110044047735181105008610693320965040787677086339902943033603259\ 658431179105110507865657714267289754654274246445211102939802826314943487\ 951570491562252476664210236562150097450415024397873391532390529951414795\ 380613953677582976563534275926393259620320587206185677183031584589863192\ 705952697457953040102524319084970004867815713198738724318893109949947758\ 061894206290733839376329332760908634933458016075572089166800771715249795\ 295996609509729622140795040297965824515082498843436827503170998112435239\ / 4605605554169316 / 220548595873384037193094704671544640923836711491151\ / 432711532962503838287560679949670018043982557768133082734012311699989782\ 582071853596811912374704123698518890598088538676528976876122948196892389\ 433086345852860801653013645073263625505070229580677316395457838948956567\ 197053293867012063748927963942802707571304305087602552524103310712631565\ 300493609802682429481536528428697683012330135114659717554841938165800330\ 017627179999118323540896623191709811892349300797762733336802714876701382\ 526769102051589862118015440394100477748153624382230624595366272508979722\ 983886294126488910972449347592788372616039496060869730917816529123763754\ 40973425017758867037960152683376076590095 and its differene from c is 0.49071121056498965970753398613282950559799263295530988273491143248145998459\ 084655801722440751777968803889978622680823845433812534421609059484425541\ 238595016845732178942227106703614228618490343907972677254752680519613299\ 957661410622556327925140890381223130356780291635686037220978105759686411\ 560246969170855505045013053974276529822342476260718860602924720836356158\ 908042008715060963024611147814963218170819586913644601598405231520936029\ 021230979222507902113376728914994297193700933536752256472632150878426446\ 610381365647981416445787783385933289977675120540956578176825618269955926\ 824023067962696839883997360859234945097161801276788566432100484651799632\ 789414237424805593571081019600338135471892559196620793839463944391555689\ 469740566347235313817474576093791827590652162478104468000354947203003306\ 475414726641774154385473630765813394305191757996836135065556135964951221\ 770293108585591145765103494310111761855153405064302142938537029124633097\ 552039031660611158056197188790230313360726498638595110947815875791577305\ 508459909208168351914005499602682321973934043990455095390869344216240864\ 337273690840477679868827668142253384046200534302899381203187641595909121\ 650648478281616125978155666344311994573873510978498974640820032690412802\ 021550852573779107979952922596790585425716029346622469406838365149546602\ 782050294332369002244606238320029148092786448689331517599276640762663024\ 111308285320665987723632234246811651752936543272465600924397195926041757\ 380662994769072962714603691305502851601316055296000678137549456363883619\ 702261517294501219396930786679353376005772558734010219436505829353712292\ 444818026282650006548309836831170325836967111217858878089177206868872976\ 227322744563307149796838184185607955196922874231932794921058646136208105\ 600006992890503507436466357652132810977008915535428489614759139529431369\ 929115112725688069071782341782732803002635791468538444678575612864094297\ 711839423110309659519878638353848042339608989958578919901926973764530233\ 826623741002655343581812788943047789936102016951095797728987382312573021\ 625791756582563756624126630699379945998085011288867625947713961877635634\ 547776946945275687118704968374623742249056739446399539046582487830865534\ 805430147230707680295321731598366581501051096399195604673936714855339061\ 097517992890641740107715502479588717226934963695178413203171762716722151\ 021857193741539426346804593005626003710954187183466991866718545798550981\ 705264263162999961803811192330505934304165867028767037372329031092780918\ 586551265001022990091975121486926497618389816738537544058200237521596535\ 351368619332185846232064144402674608924307748579596940462642234996266686\ 264680367790615162695947045737088038139642063293193541699420963979228133\ 467982256299927181487336031427516733034659314897111860132711251208622780\ 945605683121734850273251278484471315100148632753230696713403579617044945\ 265465011315843567562922360501351973326390644889134689115594405781537826\ 051502022882407422847588156783745086020204574071955902126330866511696535\ 548219474627169065554690656900807962295886418843126851781600880543900854\ 898250903602177189847483247167244951489797530731031057099106684193484680\ 798286984087688248408055457520638382613865354251567006886019515915482827\ 265523035064120136470643456397459996524234588392401991856337481015555608\ 068359218152100412005555308428783455473181139215889625293085339037658774\ 515722936666258680825205762775242804579733131908933633183051304749109134\ 275161525102470543976252100007035303832760067958706390634283957820585803\ 442252680385049814735128610351764322508433273721578489215493480265641527\ 822763056196667284928477125880019509943531682761287761318254179489778766\ 154815106581666448259700857180226181378871229266068011386503457206795041\ 739015784757403159699318007407795939984272181043234566302197574285917876\ 084259084770248724747368698972063295175254066291365899506491175128395331\ 861230078925207304527757860116524785290521872370340171908899881222073009\ 646420731453163990297398462718104050416618452817547711692855141757689904\ 106974949842422762026690522497624474321947311940325067204669748611605055\ 425978306215754145424622636245777838533581634914752331934417008755282499\ 184893254839886117490523601701444397038826296838217958768050682013870468\ 898563612252125394898566634439601220250363500143829936562669667348274501\ 681408428535860078599385977523153685126640902540318778575713579371070379\ 510326362897584159496578753790413940682706538346743097906940831057157582\ 524782810923399302242399729235104696666921657722846387296555640627688824\ 503414760650261904745539268340178520336878286384833634113055310812002341\ 371184661061993683673459651059232755182277208079714072960803810724070432\ 530255862161378099624776131963466907058840631296422500072204227501991205\ 295692523770487879211765052634223281903777691977493355683639226744661930\ 517530026528382944179668523971268523149507653108671779632301666979443559\ 621521763415997775149544675347874624703922829081284629362905879961464855\ 087904809598418561699768408542080905502511001870626050706986203296052656\ 245949767180237484660031099540297963177324923208858996493030158045788931\ 313126152325358443172380856269343458418882720473152978523226696190284888\ 136936108316086203857113042980890235434077810165347192642808256521264722\ 772828574896639415108133445172006436185325521733251720939508338979438543\ 638650911006391508157240459995260190246887274771198462315742425427098209\ 765594934112571855357462508452480237107435578666052420923996938596520986\ 717999611788621154611708974454180214424893081165887734127454172773863910\ 426290954562460660231549988295083595622220444528150740348054536447311402\ 157400971030521382495531523428340144292880097547718021581340277346231655\ 098387554493611807931210728193874897117529450874005963866610413337010073\ 020141157452042162568559764888336577843469538310088900292266202734800244\ 757881494723378470604632998885374507107403279651535387135499345856732067\ 303298868940094493039142693549462137747242264148305749474916887712466878\ 622957738615757504002465967499140838618386085178745783432375904380785917\ 292462269795649354420298236136824342850689039845627913721048000657892523\ 466457910352490067727473365641361382258658958500586622006696525273166724\ 423932615608740014607402932677045353076997128474482895231146604728444178\ -765 38143748258588174672806088855824179497627 10 The smallest empirical delta from, 100, to , 200, is 0.1396349970 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |--------------------| | 1/2 | \ -8400 + 820 105 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |105 (10 + 4 6 )| |--------------------| | 1/2 | \ -8400 + 820 105 / B(n) d(2 n) But , B1(n) = -----------, hence n 5 1/2 n B1(n), is of the order , (5 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(5) That in floating-point is, 0.130924247 It follows that an irrationality measure for c is 1/2 2 ln(41 + 4 105 ) --------------------------------------- 1/2 -ln(20) + ln(41 + 4 105 ) - 2 + ln(5) that equals, approximately 8.638004594 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 17, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 21 1/2 1/2 21 that happens to be equal to, 21 arctan(-----), alias, 21 0.98456562311807832437 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(43 + 2 462 ) - ------------------------------, that equals, 5.0585439651656041513 1/2 -ln(43 + 2 462 ) + 2 + ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 21 / 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) 2 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 2 42 (4 n - 3) (86 n - 129 n + 27) E(n - 1) E(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 441 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 672 c - 1323/2 and in Maple format E(n) = 42*(4*n-3)*(86*n^2-129*n+27)/(4*n-5)/(2*n-1)/n*E(n-1)-441*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 672*c-1323/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 42 (4 n - 3) (86 n - 129 n + 27) B(n - 1) B(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 441 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 672 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 42*(4*n-3)*(86*n^2-129*n+27)/(4*n-5)/(2*n-1)/n*B(n-1)-441*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 672 and 2 42 (4 n - 3) (86 n - 129 n + 27) A(n - 1) A(n) = ------------------------------------------ (4 n - 5) (2 n - 1) n 441 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1323/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 42*(4*n-3)*(86*n^2-129*n+27)/(4*n-5)/(2*n-1)/n*A(n-1)-441*(4*n-1)*(2*n-3 )*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1323/2 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 2119575890315318700898756797881528534578368027624172020180373938905405235851\ 421034134313331556260604917523805794301732834435003842395192915049954782\ 461153436008823390715738480419021978706908374162085379887838646405599403\ 413122057591964008620661845356359176268187986142650619599822287016241427\ 606169014139776392769031680711214137484488863682254391468972178019316879\ 659271991695970991255908109325985649467990242839772412496289168924130518\ 912451448006254014533819357871142917264618772149190414892292174395366015\ 840875867043780626975479498025156507546142532619447126281881863875168157\ / 7713650167941062711691588467 / 215280306416012183106644698005096115158\ / 980162492345348422996668512282030872383778509408847232065558056512913360\ 417727006790758925018994383613433052194003478603209200274087111656893136\ 981863616686528967797579414315290380668608058666571172372159072132044839\ 474445151375376051315388475128413461664938822502631181684486816644659179\ 253407825433666408258954706519909365469267762537547884490336971894829201\ 982570049497641813347776435658781970816560103148440076244890545010268887\ 084238982002474305837746938609815794570115574258859855205862363012172678\ 85514640209375687155977975542543731902973588251354381231535337125 and its differene from c is 0.25994839460450520526850014429216049605938150423692000425070415846833666379\ 157521643915633054398325277657065517340501259951766332944257150978314073\ 667524675236876402354000474865799020901903991050287609477278791328528262\ 156423859533453214848925311883782549641298594754300121414391681721719875\ 769702403423219407938194585519271795660266675178935247678417284165707251\ 700884510831487195032281982529740481400260204148207258263155371701918749\ 150974558922735021930525543280283628480712998522581524230533429372125469\ 089324685488741556941159698680367883090500870785269092444658359734404369\ 430181960682007289234628890496511387046962295187636114647901104149610430\ 089343387210071494070503982697861184097478585841681902822922590793070519\ 464872876893824276047361279302259346767349742713444393236773613199046589\ 663544032066861005750372190448064569326410069682482858320030038620031129\ 860998975926905335234766659541076285850746839733079947378379201474312204\ 569175014844378008338328288352358907890947243204988959988095891655324568\ 654421436532124250532543218349387911677597837207747582293970332937160436\ 708527227677722161399438082119619371420262980607785540999803484609662738\ 630719196938949831446956637764718631512279503623650227647587292211192006\ 232068200838588223524093886005746201153421115377715340069446765628724454\ 760533163005312603783934892519536800843281744788958696002447945346232769\ 843159203734865591527133989581309874606204696900859778029275695117889463\ 122710578446417504738181184792555636861488965678335651977683628132794477\ 757018556232743574672014702218711325025038299690203633439104030594662148\ 286192538988676631688568754641681793411332641406626269620607541265939956\ 132457838286111229462530606916931233159291281042483318707807927080559010\ 824895387393972753004491057758324304117802125859308048769137284931758962\ 884368354239899301453631869312567720972893226905412589944977022026557890\ 523496825747715972182642268732508282814597193551825322398542569747432986\ 410464334671426629228339633899713572504559006105058344540029393225602203\ 239577741963427041327059557841257288360277071514299835151673774258843198\ 998337476135432868624897590945200049878814178959880548588292145966739589\ 376403569688636219814150376002919711599981688620644170226882493701658423\ 546634234772420350367270591303511993685083165921933474210494942862487157\ 167562765332955861924851141588280773965956884520943145579856263861294178\ 179862389714876349689320455226355250674740081091358728382589188895624036\ 708375609436284338739865392276627840440758154463295234131676192376705261\ 898384651051914058006315587116197678483174792904501440467118110557685487\ 836043871424919651526495899045796031931984160300518521261106356606272565\ 689055792395401991175462702760456071102954046819926544398884088864627943\ 835155251298703814846284337279063838805110271606164008946643569372055490\ 981639020428513267604288712291540027032162837664770851132540779015746116\ 779916080653273180071792926861158653590506290913014482714745339316772998\ 766820489556226203099233568413417963086638171690012187635883943933368692\ 729496654415236887455340483687973547215888231274628463989794130458271706\ 369058192786231753120023628553566047782623783609613385376461973695695450\ 469808599181038403227397748143708628031503493645895538178543897543551327\ 558347587173252269347167315500635436395268243829274147461659803583473666\ 539669231757433776450645811766730661031505537929851985934568884699422293\ 815444539139999640749986088811497627121362167204768456217428645834686072\ 113574055485441833666713010572548617094665057031757860549604436999069894\ 635763295631567774984960934770060010849850006687682974820661350757745216\ 609842682099688514844661164059106113113620165856930586541320827121339150\ 798661285423120059736117453620010111208850192105548183902128547989356709\ 502395763041606362311673954997557587418936931138123721749617657331383977\ 711515632861895544831563508284250838679577810609551767258178262712229298\ 365120824164009641016279729009796186357535134316555221796430247395592480\ 774068739416352303022493666600814203624538141833110524171629348857870521\ 092028712392951886584995987226848878847975177694415127138255101281152951\ 051617097335527845959619341953936658267338060038335247082425804371442732\ 008300957770508908797551925949558630970653230151720131116280395719252000\ 097937654789545569609511090533816945845365784292393880730551945323549354\ 805442534727114515772668202640349984949928020143276814253705712971706625\ 630671038352291330940454937693576280764216764218102262572541616961545782\ 354798722213262270137684058127983126750429428115410925519461466733225508\ 830638490216182394415746553335978375281340323295104359298812809652890740\ 044695395050966247212604243399233712744979377988785601378660634975022551\ 930704481967689689670334711953433026912604225099305333810597075684023254\ 568456250135253454466793989107259023395973821860619825863318644768733086\ 176558323521095293451179512308784755940766398518351291962832929052054357\ 924733255033717196647831618996145570796871685169940656075155233113825564\ 266014507459480363680318531079012159348663431720847129008598213773631379\ 870153313637257366264753592556522871500530774264019087657584113106557603\ 602487198933474993748282950019940568902332266749952128695075943194376639\ 181992993197853690627951564441223596403477604094684148160438481516194688\ 105384763712442542668343823438518485974811356985917878223708456423825011\ 222489709100461241726099284354475858172434131477363743176014343343808432\ 253814602583387578957908563329048459820870866105499275841552792678195835\ 292179219077929085412713442072815021932566112393632258922304599035621592\ 633048486889796939935443673961511508418301416203796810407260290684129252\ 980359898004207756398805398493201790617415654804754599228507588681886445\ 123949817203721884012987857139488359557152914377135168005440228744581366\ 116808351555467831355799606115736910705196799116197908650295035004646462\ 919903013127539113144091571179858294351954176091279223737413038497141959\ 534736803998518969575027335530858945620707483537258030118460218392196491\ 458813576446926852419638590709115561792616848873174541474071186935606426\ 667535656312456724369276314698284132508594055851701432796242933480251167\ 682374262965574038671587468578428045883164803405082379949306901831286787\ -773 450322725049373885057394561727029 10 The smallest empirical delta from, 100, to , 200, is 0.2651834861 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 21 (-21 + 462 ) (-22 + 462 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 462 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 21 (-21 + 462 ) (-22 + 462 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 21 1/2 n B1(n), is of the order , (21/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(21/2) That in floating-point is, 0.246393783 It follows that an irrationality measure for c is 1/2 2 ln(43 + 2 462 ) - ------------------------------ 1/2 -ln(43 + 2 462 ) + 2 + ln(2) that equals, approximately 5.058543961 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 18, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 22 1/2 1/2 22 that happens to be equal to, 22 arctan(-----), alias, 22 0.98524874898109689718 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(45 + 2 506 ) - -------------------------------, that equals, 8.0828439211443457500 1/2 2 ln(2) - ln(45 + 2 506 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 22 / 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) 2 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 2 11 (4 n - 3) (360 n - 540 n + 113) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 484 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 737 c - 726 and in Maple format E(n) = 11*(4*n-3)*(360*n^2-540*n+113)/(4*n-5)/(2*n-1)/n*E(n-1)-484*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 737*c-726 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 11 (4 n - 3) (360 n - 540 n + 113) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 484 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 737 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 11*(4*n-3)*(360*n^2-540*n+113)/(4*n-5)/(2*n-1)/n*B(n-1)-484*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 737 and 2 11 (4 n - 3) (360 n - 540 n + 113) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 484 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -726 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 11*(4*n-3)*(360*n^2-540*n+113)/(4*n-5)/(2*n-1)/n*A(n-1)-484*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -726 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 2884159145034132938534568472220469338515307394058130727551028305943161458112\ 621989369207070284306001447034696001443757691984998184336903539863752519\ 507581343663975982983634278420516702152031721016929431461327802248478117\ 633468422012505809563097420244692704128474901629688478116574122962444252\ 399043700757970430612363180392525612354587772707990294591009005698985001\ 302260757297666967900781183890846122637355696514428169435501726744034244\ 805184704235149418816586928075971240521821489973221521833120814729818456\ 569034214323407752543099693945260134503636218715737507452384573289494101\ 126674372957966155109381773029394987252997963477498857716886346560135035\ / 375448295054933184470254 / 2927341088244780677318101640173000096420353\ / 078724670954497248718539283329682228426173410370688884489839814442451402\ 146331005313000692884761534504828195707551298379177410218518755272509440\ 237456653999654367878466718872439479404490013107789942045246314750629344\ 856489545968770376123952491952737863765806059481716073407797602418374929\ 225988366839754346477906715513810901500721133157140104621875041447663165\ 342435288982685673761911056187819839516237023751180043975086587847118038\ 669469472859507440341337364334889452826059323784975113884242349965598743\ 485922602686565513353050206273589947305014214543158786609448315047318891\ 143338871718326632152521766181082437290364627868691872625 and its differene from c is 0.32767889730092383234434155947618992682332687280308824193259061564158950110\ 439637198902213255775662377604539430861683570500114516452843910577021019\ 182374234566333424861612121337103327496960228852645216111583665827879957\ 283196789124840455995215102184564717418338164990978913582070773425799894\ 416444920521464677755970920796063024181273888453378003423889442892722799\ 451739415437209855887683287453944163532897760410358920946697494989342793\ 268037238368285911640538330702305907878660071677551064267649418146971402\ 803844787570129864368560006807047628359117387383083189772065108015077515\ 828978895377318990659500136493577363608324562803852778016161768700700815\ 380516963372035840493545836769968761620475325440313884035951972094504237\ 882507896300399689715166272218450721981753317244411794847947630570270780\ 541477432073617658108321329237443848634098894453410759127227887615149883\ 693578596550253201271456465466025960961190843357655855388965531252211427\ 527677258871092289776777769445190576772134474838407627535450843255661208\ 689071199716794654569926684029863450060594693468171184952792144300431254\ 876153554603980012942623470125769678595275127983122187719396190917912368\ 868350288012191219029389058867191542170406019204080489057825490737581656\ 092360846276242505640746026526067220399719047027914267552087255020418204\ 566346896798956773477973457108156314654518058053810944289995441769042148\ 071269813801320204405230815494104885715350690293606893366566768439611369\ 684641992789554375470795477310273904802103158969895850254883510115761512\ 297078387640042763906049924319241008005288764703629825695588895426230966\ 673692357229670234705173572577553495343525545114973706520912912708277694\ 902320859572942279923257069895546047795598689631877178438427120214264614\ 580685395596418687135467839127605198568027104198751069248951474963273055\ 605053483887113939243640212948835937699469390052531838207025557178683193\ 122707120823333488489827316619747326199864443299340511832214357710070723\ 535101497519506084289435034408200076016723791656010835023423563473996954\ 168340131790803046675598698710102194434271283935541280115955884043999989\ 230041121247658002623029702854000046139101412220595097778939082281107071\ 699079926987566256337974223551866864065752868949791895068805234152631038\ 560720529334040677557492966074268802319888923646912950900742487985631765\ 740940729825462792254004918367521633871345200044574531254849104332299817\ 252631513324039055524431716661457157996648282669139850574987503588342132\ 813610079960668821610040035122215256566738361460696142912023288172731688\ 325514357247010721686091719056178344063627154801607372625560383262475127\ 649607158762677433789155720103371221706908594145395816284368353904792173\ 654990729609872137947269548057782046245851034880932240022642541530767523\ 317246913879849010483130926862272077326880931464406305641475586911203215\ 374711465268995978930127701173730859275237415221896966809814737330609214\ 532252933384458349489371821219673945247501705633994569037212089654334092\ 069467507399689910300879540604718931448299751853477194535995754528883315\ 321033296782107181873733073124644068128632996347096634446751781224284009\ 397954594645302473045214777373353264200206370902369243828868295862947934\ 866339053720201424207025549538084397667006229445241779912812396264478971\ 057431371576309765194785159281871616766301280552251386708279597096270149\ 390369391948801395597727474316923977022696391894149128675123478786259517\ 899523998979777860885513057983196052962433947922646377743215703576887642\ 222014885114138571800374192519363439693895311627762428446299091151180349\ 653152313662457335065592182762333096452503591304864385568478116369357272\ 785785755957843281617866781376347048634516419830231205573159477949499333\ 667151530301291869316020301441400568949179766331749593532347277550426525\ 598351506341921891048383318178546312155930501535312683617557568140899649\ 365076874111498749298915240190542590960195163850271220173338797437610796\ 682656723194467628637408510236063669633148193662066697808924223604687683\ 160168754808855938355661577055295510017899159425052165860637768658968360\ 659826803152838691572120606072488150545125450119279359767391099529924693\ 441735380881436469873596559682857587433849039575751270642716257360853847\ 438266860807052584489460945439662779135450105451465407103632827271162696\ 350887323313018529195448542732845710333310417981463258708847246034857643\ 064887841640049433887777296064510614016633393170987389207095454761400551\ 050267827762883885585548330442034386485177805420858264661213615155487422\ 411070380754226568554313120745382363493568012978449833102379186600914398\ 238017561151160573103714344639141515792096595927242850999381388530731814\ 335819576427541068622630970408114263526065056363109109325831713459629512\ 823655777028048525166113015624955136978470964673366216240175294333558049\ 575389591056532983016022976805964698001946301709300091684913070959142512\ 356558102302948053107382202539154544403089893335826660736942146212684193\ 386486842705584377419426396217739457675648317541588639668245988844937460\ 157545845242291679188657520286084132501691484408972667132953559108341426\ 084120479474909538265207598865216079699426867988945188903086804704901037\ 769635694928196850267793700142696929865539865313107701187285663855210123\ 900681632908123825945294674458122145980085119259814493246698282263583638\ 448838514492080319773248772592331903422951535662917678276228398012126785\ 420742656241764275871785195221062009477992332647369755941504224288838399\ 604452537462362466289665069718401482746888554523757567497105487562475079\ 986613636374086684188441173681354377162759866368742730543297583483187670\ 681761609776331265895691200873420267961808629752752143369863398486629127\ 345014115827568739496617720169022225842237832576166174718946595879363790\ 300434485629132032118546462024360748892137569304334821550614780114890115\ 919902110978457323978850561100359945345565049010965474529407799345785230\ 262608405497688269295017939636097887407283110139167543021248760369394070\ 546576705921279453035886814033912541609751315039888182293924580151982243\ 141642930963369050441581465789978057295593422341141801919579661624385691\ 086335295219710432065860201798346885998696724157545201169047431440327852\ 369849500017015034127397274061960542409906537120761594078645412808969055\ -781 8441445620731013683673185 10 The smallest empirical delta from, 100, to , 200, is 0.1511342391 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 22 (-22 + 506 ) (-23 + 506 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 506 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 22 (-22 + 506 ) (-23 + 506 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 11 1/2 n B1(n), is of the order , (11/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(11/2) That in floating-point is, 0.141186226 It follows that an irrationality measure for c is 1/2 2 ln(45 + 2 506 ) - ------------------------------- 1/2 2 ln(2) - ln(45 + 2 506 ) + 2 that equals, approximately 8.082843903 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 19, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 23 1/2 1/2 23 that happens to be equal to, 23 arctan(-----), alias, 23 0.98587396027645998092 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(47 + 4 138 ) ---------------------, that equals, 3.5728330278283836854 1/2 ln(47 + 4 138 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 23 / 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) 2 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 2 23 (4 n - 3) (188 n - 282 n + 59) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 529 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 805 c - 1587/2 and in Maple format E(n) = 23*(4*n-3)*(188*n^2-282*n+59)/(4*n-5)/(2*n-1)/n*E(n-1)-529*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 805*c-1587/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 23 (4 n - 3) (188 n - 282 n + 59) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 529 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 805 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 23*(4*n-3)*(188*n^2-282*n+59)/(4*n-5)/(2*n-1)/n*B(n-1)-529*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 805 and 2 23 (4 n - 3) (188 n - 282 n + 59) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 529 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1587/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 23*(4*n-3)*(188*n^2-282*n+59)/(4*n-5)/(2*n-1)/n*A(n-1)-529*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1587/2 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 5019724388064634984405769489309815459807298221680349782794298461397646418257\ 340512407159303609206160217656338184927418001441108459882948297545836177\ 140999806228015013505891793130961712524851374951031468390563483954304676\ 894874331974976503995518537257519927643439588500559045429516927997557219\ 869075295142635652083940865992241746426691682988574700763213831725252869\ 556932866032782481812206442258931983576762860678715055660122698570220885\ 224107200743373917327707537370324798448618313122498845314931308206218756\ / 8691851482229618647416101345418257254326506554418208049 / 509164922730\ / 791848509140875645541779041362971233739486150478864595344371612235132684\ 595738481630177183496013107068786638272764613439188576148407042340984917\ 778542780381097626572396324722067515396081155951355451819562604889456244\ 419224974465711229839288219110759697457902852323696157695931667166877876\ 767605647995980876306180140084814486248405114613282594504080162784160527\ 401479777062345483723308333400715171258728445098494150716757673913245133\ 367629584053354289381729412021496179581130017585130813395693789844325040\ 10772703528046389950177880337657169641909152000 and its differene from c is 0.91144650664498786442780771003887289781175004072636075279326326248481688278\ 693584743736044435627569408257390343839471496610005405827388742806615312\ 639176649109322830530582166868936884720827754362308296556691770299948138\ 158568641139243307495984471214783463865673899457122794517576864462228728\ 218476643055496244945166391184014703364895296459694723159627776989573001\ 806089443563349148225739216644623422653291430964306080030997000344085344\ 676470205450648904932234846837512658340123641406588617238844934906383288\ 556908749297485150005882557737213653631001060378937589339721997749432229\ 827592228633466891597481097728895597806196021729568385951230806239550433\ 050518400446058473916414322970201755295446151298966376229080906787296549\ 147910253494598552482570471512080883576255468289715354777605426102664458\ 463009534670807296936380599481059982300824300397551557858759361687936346\ 955037107416692902724147936746938010386549515792513609533039633296098237\ 730917115979537258566478134218026355675914707086791837136022908187889366\ 344767800966750994212151918147399538617679975227427350779349471749398525\ 355418112741032797217446268195543713098182010323606508308591685327501437\ 500955457441861769432933135696177994088867598123690637563108728907316636\ 433637016562030038697902031083392944727516672870206790121493349336539058\ 944635980168581167687114639524977247598919371410266727292636434585711811\ 907549413510247691537323075421756645290406320354406344639997026737731339\ 456948648069930254330451592103798913082181539091717641420027362363729305\ 246038521246357616889501523037854205748847157743798942739676852045371524\ 251381518757314306356341839865844589421442772879576833885706479859975694\ 729594986665946909384117460939387901647681536207319975087102729948320466\ 473793605046065224884253920661236161510798016485560810773883481745439952\ 997043802817721221834439462175803932081020432214056477647540985796490112\ 474008909209434796757210989538587693332090540808091504379993722167159441\ 253456935591167795120919574126494947096453350005863317743334492311056184\ 208024164733745230168141203644623690843669011027139003430351778455204601\ 555198038031003569479874546443197961658592536183910123505945476825193855\ 224725465442730059890593685604879100086300173680363861985776648570010358\ 691129254079508559802510042925516150516199046192698709488808915429777820\ 905123006091456875886608164708387805604041637668139288158707257600733849\ 019251859732317828601373783246897062738890442504740236572135040727660643\ 255270768640039858541208312913952144314008740368107518367841608619241142\ 622438553453819721265959391544822770903761307262434335251176126568958120\ 599851054687629847704207631653101629348284948683692379016190072526559403\ 274209712026783570611345497921033141029104527772130346849444887801296556\ 063634833127173156477980545743380159265491812861939417277813399377975795\ 900207944128339560030076940779441380878647427626030899497987741949870872\ 843348814834904359435349723499699424349180557504283940349762566316417850\ 103585240665046668283097417721331920865365490468902714265533345409358826\ 393130722326532177555749401369335440181969687115546327015314573998681237\ 090984219829122235678802697070655796328223346134734318036567086494072950\ 785206971500643732423471023475432691974863743846760474629926988683280199\ 485278394179043433848711543509749748111781713149583646557659865976592483\ 916554609316903363097411548924478084809115321134878107477369665883246329\ 543642963974384977618834331486370077628911409350519745030775152425453237\ 424227889431027247146559092555661464575907763438573332718081091681292840\ 692662123225423324692529214890377442888103881744849572282735532519158882\ 931805812870753242299346068372021178335964910619448455171435800210413367\ 951327086547323730153940153317414621311625385675853671427249815480235190\ 282809839184064308942183593708005068836844409778172597764978446289970967\ 254863973661750108615344104391509573165620129458302587477526283899770606\ 729924808625321502781154061465022329373166522397762974505390072677789741\ 711028658609819122034326028894892465523800106372575804327518853943782607\ 873291305769414221865231661559075734242566772762270784907460459907779163\ 526471619546164205326925813463470307353098816870634618222933421991996704\ 255995291708305575484470596375620399275949999070556202903158148107265957\ 265444232101104980922276883217056490846664753727386334906887420546919094\ 127999606369157651517860067073833328057460053703978675361420236747851990\ 705896977376965362069570884374332048113683131119291732134299484042909155\ 951368517484284683192085163372982107567357475946821174436017321533890203\ 549357496430829770235869111320374097194632703898865675183396236488298706\ 228843003312172012633884124556743566810611847031689492215792696256849632\ 566892587705222606734577576584875437132377154657255725698549568003878870\ 925216012458599298367166214532934463958012541560979153086397132637434672\ 430418081980039363195818458604560590427109168384562742671121763795700239\ 320965101011246547078780309889693194643054357316387948649689679441448244\ 135985395005571720754517151112630052462191793946797558236744278666431481\ 440803918598085468476665368790833804684478995172075240907133457735719681\ 619701058277588330432286609079789103241490749987418069081358534295997427\ 976870521607365787622486742316516666145861379031908716384100693663256893\ 478534221931375054467930592481248947889715635565795610512696681050520349\ 450599009353410141213196915977240362886506311353998527736149877509748227\ 069681602872319933049926952524099362606104394586786312155677999233527245\ 515724877099696601094212434784499725297566863298666551044465406149626482\ 994351829884546951171886278293582428318786461195107456172315777418818018\ 512605077456445898832726894589578335737538450347016035942012480443300514\ 510524154981522954555571230844014197226577506936322847629599897689349672\ 126276518797468196486609160513105185744343652162652006163057442071265629\ 230177153950201395697652168843631512539586107948138790949008108823212818\ 646946991214211066899336126519060853975608936770744123407237947878739400\ 642828248190866409248797468195642652411599258068229228581547352978804293\ 312398227834875780255145188896054589732400760931668030912175718650902730\ 771122309148205506443242540565071540165584916766883069494612429290385449\ -789 97562453497485530 10 The smallest empirical delta from, 100, to , 200, is 0.3967419227 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(2 n) B(n) d(2 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 (10 + 4 6 )| |--------------------| | 1/2| \-12696 + 1081 138 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |138 (10 + 4 6 )| |--------------------| | 1/2| \-12696 + 1081 138 / B(n) d(2 n) But , B1(n) = -----------, hence n 23 1/2 n B1(n), is of the order , (23 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(23) That in floating-point is, 0.388676844 It follows that an irrationality measure for c is 1/2 2 ln(47 + 4 138 ) --------------------- 1/2 ln(47 + 4 138 ) - 2 that equals, approximately 3.572831429 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 20, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 24 1/2 1/2 6 that happens to be equal to, 2 6 arctan(----), alias, 12 0.98644832320812583986 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(49 + 20 6 ) - ------------------------------, that equals, 7.6505623343655855894 1/2 2 ln(2) - ln(49 + 20 6 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 24 / 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) 2 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 2 12 (4 n - 3) (392 n - 588 n + 123) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 576 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 876 c - 864 and in Maple format E(n) = 12*(4*n-3)*(392*n^2-588*n+123)/(4*n-5)/(2*n-1)/n*E(n-1)-576*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 876*c-864 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 12 (4 n - 3) (392 n - 588 n + 123) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 576 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 876 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 12*(4*n-3)*(392*n^2-588*n+123)/(4*n-5)/(2*n-1)/n*B(n-1)-576*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 876 and 2 12 (4 n - 3) (392 n - 588 n + 123) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 576 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -864 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 12*(4*n-3)*(392*n^2-588*n+123)/(4*n-5)/(2*n-1)/n*A(n-1)-576*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -864 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 4428128106011152147856297722722144118279887577286539583332052136762255712000\ 696819312047212338224314194493881808851645419685395117528032531367672551\ 066224190944087082937863318108758271959038027820849580557325144534649322\ 972515738467511373154554728447019498722735737180003326801164134151588953\ 610764356021538572937040478434633042107575668182035017828277548530251057\ 606942710319184323112590055596974183300359474305750112649065997905994212\ 318414561539426188450385160921311195849608373958175638737419054378267332\ 242189181381181863860423098809691477828391722638208937102318446980882918\ 615159631750926616134984890234614562013181913487458933582071027022583422\ / 412200592518184961624575721966248 / 4488961055364765781269398323213902\ / 685777890740285331897511331305010651844151962959659039147692438621166138\ 866970319111630034069340454370399512927933712697790640554810562955511952\ 656462146694708255755511684788245496269823817086210572936738960778357693\ 201754637182301249255752175728304360549229259918876610693147001918887770\ 931284306357039065665213774428189771023873173002393990396926383828971200\ 937555551835359735687605229259676368255648800086211359617805655510570204\ 251322448518319558694047037309468482117077880445178531907928252970988162\ 289204132776323888608693788171826289735430910995566858740058206516588570\ 017695646667844783882652616244929940219085497391430130235130353581899386\ 625 and its differene from c is 0.52367286036428004891357311719429946004374061781293528070880684829867099444\ 192500765433464995639650772539859020646087031486490521840710703061030373\ 810248240865471575728578345603613873364504897622516486983536388989692742\ 144972753333675351016613868587455618518958184080152648127965744929552554\ 447529230147630920467518846383374329756839872893965085092800481852420089\ 976484968008287532932536341898872603920913383100793617263682934303593153\ 093427950964795350838412618130988594491265862387465338053046132465414354\ 075321089896359475046641978243027037301615649247062272347809637168960352\ 526320608987504171343055091631094410655146733491121017011285660425346722\ 146396889101761861312788918431919704790918279660748404570667608276696121\ 614945867516074303378524209869331717819364251488163382661661250653243347\ 236791000194177210624652798166953525942764853159959000107226597085572119\ 308117016135772483461071318752620185220413966130636073826104584697018911\ 388655020786949987238828516288431286637083269350333933773449720966727213\ 593240294222620393534743150702735012026464903734727471368337544565691796\ 932088643536286146568081984087000031450023712891410457643372064944742226\ 657998543605008785620636881176102995963441848994112412322332057138154578\ 240897738234971761057073125202323774875575387644600566486591981930645941\ 580067958810553204892773474336126964290356006336475342247692465699081385\ 280214382751026437170836794213199048761847200837649162020150252807052256\ 133831156184124584198907928156283592039984461499716158293048440892637683\ 927190950686583185048110308593656268521523572310485529573315183202311091\ 969973086172455208351189496309522083025312401670595794740631782435517017\ 261552116571724498606687426431775680742490958215920618147456741366675776\ 585282519354790661641831728677629253265922556962748794535607943328676491\ 317153134250972559253977698419631126958557019235047941631827158230342253\ 482090863120735369051114073595492782218432485120527213860352006557663449\ 354690957775639418241331444180198162277244220749483706873010173190587827\ 645404096188948790059290553997560162311992761868356454404697902104653169\ 818651922265458038342295590639710285553894896888001021478296253635376891\ 654909556144492688023309857973405375199777363149777390542113647631170289\ 181601293600788331626792417007617351473393036487898677883562893902389481\ 499605284511854773147268475856651791324004370223771249796196089967230842\ 694947278334654066883355910575609027062340112685250651874443715299044645\ 174085433895516350484935838913554096782791830271877080566403136255878849\ 378462842600044240596769675125655563471071286207036174764898654935825795\ 819989745723814482537805783259650454318742288995758913027179491927598177\ 970693250523145940610752760562044665539732683127195802024482508185759538\ 861611930436600689015643285644778425635073062480829659729917963764056741\ 532457299406055962178377791755122984296618217653575504457205396845054156\ 629001586716107172449762046113504591549815852687339037647699341690803720\ 029832152171829314823636558313302162133253020664244366359087485472617421\ 367816887673818297011416570176906994480369378649961648330397203586159768\ 618744803550235675155935357247976196038657961804354313518616355379967495\ 189946910576835619171830211019981892077234856936506450324427615699646279\ 407004108142048351759287627284417592781519618036114248022098925820103850\ 295776233192166891967624964732459069277134018340000500582732297648759123\ 639048742036644395005510631916743165077673484670510595476951484619463181\ 676484873805467969019557102486684742518050696975202794632841005496695872\ 640708407769369023139766039492325589866538556544630813815296367964870702\ 572397044877400615271392541935400670999705416783796516868115570189657742\ 583341171894044802649782500629807378907124156741020598937638687773859355\ 554825786231173115726055498374442305674541204099083106975388581963555805\ 654030914318097497915486473667163649329519906841136208897141170262715676\ 232313820238153611208375323637751658639788319831373064214068599562792244\ 349900472687449586660954082227136796462186038131622634744446179657370919\ 864712236971101738134739211580144100775030839788769915506719266147575333\ 578538103134123555229393797076637235167138363159152951215290905075173846\ 489118116896758824822756952715541869366887387479974028347576453009969882\ 058074084081750374650431863266157510020857370450883175255707206011675185\ 117640201841934338305362878128063335942392084408570701272276287089042989\ 787864597495709147201277656758647539114237034494594972455494666419295389\ 577641046184166813194598765385777954716029429813784472964466104528286775\ 712407265675039404561865362871388637241976226593740573470876851557573140\ 910741074267978493359832503220590483516935219035295103684502467856419170\ 507660478663279153471066909023091852639684028667598442191048995002205596\ 387411302852823144113727885213673011586815766424646278130211844345861801\ 132816125045581147375757995376119052396920043938478726026804884776529274\ 862557728555706517202832207443938962340481900275332406659120944951438784\ 934688907451187995282959906749821192214969248864334454585733450407684685\ 270279765988228268832593002059638220569293852169462802213734160423560502\ 346135092910241166129951566245971543664028262547355160401121354909791649\ 040124773794606667339790650272442350823333915333086732820708051381087894\ 632940465495992337959721914740630464956315540443599626231622207864763101\ 484759767243492728784924255910753183959179713792127693790280110744004425\ 490284969739276769011342473044161227863759826800474265353574434767773032\ 025438807893279954716657219274484880961721678909986562886714777030453690\ 383391622397943419158672490789960247733400672663127834537528478048077868\ 465652842342535972869060406957681197907893178396506981301289235067457180\ 604176900678462528136380999002831652730116890505946995225350214532313017\ 448279375670514200541284464007849189023716930523409760768087373299924992\ 665405494487438259062914814949848584143725044883437810956752170526918871\ 198108433039638767221761979783184329906644941731502422311065195881978509\ 390624822228188127170876846199982370124897921472508920775035964759270405\ 379679822345447481141463757266478449345428826286685004554682881621073632\ 771606915918283194116693699438199444525558382050562006683853921395359179\ -796 6212317029 10 The smallest empirical delta from, 100, to , 200, is 0.1613943145 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |------------------| | 1/2 | \-2880 + 1176 6 / Hence | A1(n) | C | c - ----- | <= --------------------- | B1(n) | / 1/2 1/2 \n |6 (10 + 4 6 )| |------------------| | 1/2 | \-2880 + 1176 6 / B(n) d(2 n) But , B1(n) = -----------, hence n 6 1/2 n B1(n), is of the order , (6 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(6) That in floating-point is, 0.150363123 It follows that an irrationality measure for c is 1/2 2 ln(49 + 20 6 ) - ------------------------------ 1/2 2 ln(2) - ln(49 + 20 6 ) + 2 that equals, approximately 7.650566841 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 21, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 25 that happens to be equal to, 5 arctan(1/5), alias, 0.98697779924940379185 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(51 + 10 26 ) -----------------------------, that equals, 4.7883274317566197203 1/2 ln(51 + 10 26 ) - 2 - ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 25 / 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) 2 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 2 50 (4 n - 3) (102 n - 153 n + 32) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 625 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 950 c - 1875/2 and in Maple format E(n) = 50*(4*n-3)*(102*n^2-153*n+32)/(4*n-5)/(2*n-1)/n*E(n-1)-625*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 950*c-1875/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 50 (4 n - 3) (102 n - 153 n + 32) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 625 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 950 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 50*(4*n-3)*(102*n^2-153*n+32)/(4*n-5)/(2*n-1)/n*B(n-1)-625*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 950 and 2 50 (4 n - 3) (102 n - 153 n + 32) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 625 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1875/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 50*(4*n-3)*(102*n^2-153*n+32)/(4*n-5)/(2*n-1)/n*A(n-1)-625*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1875/2 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 5622202771818272601421663468471608834494739897249741424378081872919044238725\ 594566475001834474615246666916789405675903070692798672887332692865445243\ 093323588693921880353378366356721108944855446035354400707823846286427083\ 787064424873011184422717080592941067142889316912259396727797839187509068\ 004752807170402480008766898487811480235877053872196909783372618583753362\ 866420717130298993823040815569168143406997865143423683961138460173445651\ 392641545728700251423946494706243056324521857895605376794206931032735873\ 728067918958009117732521826755170796409701004044264135410863840395846520\ / 9472871315957676290797105192144035657439635518 / 569638220443656937799\ / 763393152411145198405586825655092548219274867120036084656582739698470495\ 125839799393446896730884112946447094186501245035956518036579942082546681\ 393156387809437487566108485634734556695376676144041956533893010042397218\ 830915965635431506706857896252404640693193130641859598028065842909218562\ 595266644843240183635968018141651765682554134700701884630595025534288089\ 451069897412457844945369187527537640094676981295377038648033195366578234\ 863476710389787222139856343548148026511341793952555268046101867942001574\ 256534781718413195495749175080600977732157894632567174021601057390375407\ 05483575051054616878778478043 and its differene from c is 0.58641591738281638647798772396026105513904116667449424804956200956733137616\ 806740796136677536696737113102719311918935144202525152778309831499023384\ 393323286786899779438736548058915416266442551165278390573489973697306190\ 153246872848607022433828501602161912135148290755553330268164750992817349\ 305961650532536169568882319821650918070150426785258511640219167484537031\ 388812037516621760069147291508776392978667878620113790816045653190477851\ 522670857305543251405197575568288970801234762648086996825547323653606690\ 684852890256769997396821258446469150340814332513814412019826388005825346\ 221033899977917451807390788830079846934920362182537470776427191221027877\ 629597785564928094377756148398623602047890290883052154066707344690527340\ 378766075393443847648793352538043981347288522206717302890556916104745102\ 106913257524337691998943665784446689968549852549702967389809926110498408\ 294309963089944007265453667482491952083379769662866417390253897379991544\ 895124454382900303124053579424462706567943704949558735559373687566261490\ 235018879098034512928926835074336774788799862280030269086509010359572377\ 790127581771322964988897277524424233332016908274034178779391184334967731\ 676653496236891487928701384068777534440902050101104687514263221296086561\ 798816547091735283073103306192019373866775680625150541948413195722615397\ 730498350575408784579559037807866075662054276171173280889168140647561825\ 516245848971623212609251445477150684478232657712027732624539259983592799\ 182774530383068769888158453399513264625897422658875572961933217914360756\ 311732384122513117935664382535915781341028000645020794135154750864093164\ 076373447992022734526763247242658198373318620448314667753189255525138713\ 928471073866884301500705220811717197798832659285535537064832323320258760\ 241889607117698102185992478321483527730229810506789299565167479106371835\ 310285104518122353138549878398728103386545856433681912933426812724016700\ 992702920208490214066675272262581287075237661523870459952195836740907765\ 676563147079621010410504770353090497604570547716200153315258439419110536\ 835862515898339342064363382379209916946932209899434711342115343184129996\ 441270662712417266190466450634515852638181036428949088480612496718693337\ 309935389316936675340531108161127788966271661990878068243341005926162858\ 391527339111830433781179333220092379935238003525494002329746062856546172\ 915744755702724696529099996943105711620571486194578642635513310373607974\ 451400792017710009691126537110555722201605468739860159292505322304195430\ 404953668422762973269128508085016825140710063566637051893092067445763015\ 326568631297746396339674855939899644858171057230823835456453868485153774\ 877501703144256264214242304434252678383314205659929125674981766436213336\ 139868020705034654810059826420443174394459335727811641292474437449900895\ 596523832542360911980073450614473497516190885000356995713120017391144162\ 323982501982163961294094156516058386725320396409270063889840281137119007\ 278857153001075916570866027164243285899819634611953350652816860230054704\ 399027219942356640622611965079515971669667579912503616384088802611210882\ 976055506118857577022356828436303185793992010385222562240983455420721819\ 254025279115970754085209406843776561905952879824216637506877041774446311\ 765665046640282900628474229356927700108479268429327176912345078394233499\ 008064157728082837391110819210618457973425226471478091988898250937569685\ 606180920381944473135777450077824233736930393432909101836307081815629283\ 047815168518931799976133782051158830392583947505094538629491309026331174\ 332335604301567946128905328982608999321545534456431787248140012816052057\ 863614250518148548915713150700469765822779637993777332169633794527111653\ 930503039988207577739292647925348174932379331519549444839419376445078973\ 503594009515223943085692596491236950485077705276219493682872806391197184\ 815975880081787228191135351720142590753734590072885155371039363628047000\ 062576248978221401491213156715623170911769346293974952461882668960488699\ 382057524696368515552574346399595423200411643755529786839080516720467275\ 725860167686174828265362072450656741086567639333203667695529341208023171\ 237666523717807237555924398052248512649308387642125500118567820953340364\ 731383130782973345926490912260460895787632313053702661044094649154762470\ 328912601327987578013286166944252484450741943613093905677074455361823927\ 698622107822306561947554251235886642885829914420052594353145902526896135\ 853723368043008485425745356296932852625408036314148191753417948396976294\ 763908054554581615155082059956236944499817802926338543655568779565101885\ 433844787260317900580022543665882954163009364515798083121965893506867206\ 862331183533969561102063700539528698160785094234176903115367408787221785\ 538836699835423763209081557537130909266788367470189992835285866188149029\ 044731661013777514198330080467533544959141327519013462760140286512201895\ 174897982263665349091356030711812691451435311361894081176640518223999790\ 562272038084276828748882468425630775139084101046368717856753010997096577\ 986152636521984361138049043828020800606358819981610147326779524567957880\ 847756383753925120385417116200353605133813294066369714799327081305949336\ 308848759499667451230265217134362866241050827225250190798312623176438875\ 702522897496701994822025728162061325356688394158501274312194941701109807\ 230267084014855549183372501107442815181436928221831139636033207058226490\ 279851006030655414369294047793534401931109809747411810231899572272525542\ 769048329015759669686569054612544885704251997333808956427401750893432422\ 748277549328393103684323100605513500538588719474114376804282300712906382\ 512254314867882536367253264040172406151609982558958010673657547307648083\ 454338190457999021553934792491672376907282950336502918540762583920317472\ 473495101821066159081419593550585282856534023482980315326937626164210810\ 666961815182731791891956311677861202029700599498359716568211341235309016\ 729395492883353146262163153995276419945258390725304408224414223559829253\ 461556340234094387815303679698231261936977087525588523052435190451373450\ 236165477483702463701526759203300149702547180783232631493351542297931084\ 126396968143407406949839917364594873653097268397625759436526695248333622\ 776420740228261572074898163432847430085589645784091966593808222611280448\ 381980816550515013360139305544549430477239881846242380844711641322299042\ -803 116 10 The smallest empirical delta from, 100, to , 200, is 0.2782920912 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 )| |-------------------| | 1/2 | \-6500 + 1275 26 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |26 (10 + 4 6 )| |-------------------| | 1/2 | \-6500 + 1275 26 / trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 25 1/2 n B1(n), is of the order , (25/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(25/2) That in floating-point is, 0.263968716 It follows that an irrationality measure for c is 1/2 2 ln(51 + 10 26 ) ----------------------------- 1/2 ln(51 + 10 26 ) - 2 - ln(2) that equals, approximately 4.788327705 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 22, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 26 1/2 1/2 26 that happens to be equal to, 26 arctan(-----), alias, 26 0.98746745293483593805 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(53 + 6 78 ) -------------------------------, that equals, 7.3032836456607769630 1/2 -2 ln(2) + ln(53 + 6 78 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 26 / 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) 2 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 2 13 (4 n - 3) (424 n - 636 n + 133) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 676 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1027 c - 1014 and in Maple format E(n) = 13*(4*n-3)*(424*n^2-636*n+133)/(4*n-5)/(2*n-1)/n*E(n-1)-676*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1027*c-1014 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 13 (4 n - 3) (424 n - 636 n + 133) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 676 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1027 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 13*(4*n-3)*(424*n^2-636*n+133)/(4*n-5)/(2*n-1)/n*B(n-1)-676*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1027 and 2 13 (4 n - 3) (424 n - 636 n + 133) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 676 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1014 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 13*(4*n-3)*(424*n^2-636*n+133)/(4*n-5)/(2*n-1)/n*A(n-1)-676*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1014 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 1395419269689411667814342283618249816515487487541428394339885581057009662338\ 456115909809733306953726115321605625496915374174977765472298460624897062\ 475419298260945335226613940796082191201624328372128456298419749797083533\ 669385424195523724895146737934852291974381025008332110767795621120385488\ 576562915703293970386336125449912475451529293478357663689000378442730753\ 746373761606701691356864441016040951753793235718569344537690273340866305\ 395849336128180739195510913842389625214140234254270037654636294007535119\ 579621909252537453059976841417116874830156345416087844641583964039325654\ 993480821108685729072327676294074655891293631272689480621451304854867094\ / 9396782220296114148897419361962325436618 / 141312938015537502306321784\ / 970266545036219677262626314673239162853654513831805547108207287258640664\ 499409494085553060629671323668196442958018192113973784418800321370902226\ 953850728749717615161180106439209446986230329375342987266157577623443849\ 311572271764368268122356522409380998343554870682011357743060402245526795\ 190108477809447853265683639912241247721463731413265491792470433910370051\ 901407314107820240973520979969699677839326680093028531483667988204028269\ 903707182481154764472456252748531208112457752554558623505971211974386044\ 048489318340181045335888800158094284468475523546193120511065510305820109\ 778382885253995048151309710972276679140019881617811465571874468706273230\ 68050650142911625 and its differene from c is 0.12157763625040798512804999136535159976358527336338609157037163604566882530\ 859388717276159736034713951732391293667850417129384207188277693840298056\ 131931552646226943656122221074430850519249996357716203297060992196553012\ 826087073299086382566587018140843786339156683959928673047907706553040232\ 790107948081194588931476585406757004944250060265263808862723360210585040\ 124484339845520481415101283373451499843761309240718503552512770661862377\ 587981784523038491111183929633085102681499047733319366713292680776128781\ 744937735452671485043867265205155089012203018157078868216081376713900559\ 067443821175290797000724672818076437413216591825609617742652854745664827\ 647986483767868161781094957848489984495429309495346617178462813012633074\ 894992563676869857458908869773736489723895385809434393413961049991941761\ 387235374956270426745110509204686961328191545982711804384623516626165450\ 950302895154663481158540995876841841220973354088936774271125798560018957\ 807303455496480605414862509287716044509070691857051254154859053350891101\ 430276368308319244108387925198888460678855849291196532742323071391923513\ 071661538611951286197582863236472782673681069261246334708008309442544582\ 295848651127283968821819495524289727932303877168679269634439564715038793\ 728455407659605897083643911881329840624836934327227419689987030491715137\ 239245117226086564422534205414611328443064696595743231279314772893820278\ 000459221268129652940540337046794731377868822857924116535691735871794348\ 876481970646651115698471490069674965666805866218157954412679806682970853\ 746186065338067515247186841858049595377466755608524753722385177328973983\ 571008709004487403644636446946159781083762709215465084339987440883314515\ 769106288610622580275460675325145472886880185052917584945563333639117362\ 150168739317143804649217188713180311632617309228576576683325539581636106\ 223395585868496022440294407442371147888715376644841824345607563020011472\ 812815190638038552079178133267547785141836252543158515570007254157682868\ 529555556021883755795529598313252949647491280977494961798087650726262041\ 976689435418063045259992539749653084136648902367989587107670034189307310\ 758573589120657757694042597127436522562982655433473017702958391054367831\ 207080528330571402878905863643392354881873421560297222926691328959064136\ 632991149558179739881028028263399133051019065163714337155532890688131028\ 113370942897847716482238336266312862198566766789222156195842829562587464\ 336143353331348811225362183963120204221284701355767835433847970169701625\ 601051753127451861602681512544856030309413762786513320516791499261987571\ 728192710952741205934106398927756146542187733989106728114234253031603687\ 505539957196810930241956734370421877753273998579794294179775680219521987\ 203516696375177312996670016516007682772886357545747604191720279762345246\ 040885943009434549989837740797617982840969856291520332051023823024661030\ 478809746121588690610738142522206256211838540265168247254634783033432113\ 356599452810517312249845657355195361654841450961629222407726624289813429\ 761291274930047895463461677820796471817013420534118156494193908954010408\ 047140207553605613571433075621317354312151098825716523804832880542954135\ 958045786743654811834109506180913524084104899376284975319718485066516232\ 679376257473120517928590917857625897500801178795483063323834416399141940\ 542786293983619621295675751412722249276174403860150429576882002240214779\ 827888200470263219101579911476386473257342250513926688077161709493314664\ 480385688710609056021521855089665539613480813800778261353718607888596455\ 236542923062263710355553184599176694525119934675255740243895583740761624\ 553011672427232129520845304857500694542661812246063112812163359472111178\ 988047161697487278283987561801739124945491441731028710852520376564998896\ 816025784828076644332628958535331422431759901375440364032625945221208882\ 223633875131706137904156767048274197114551848740072953281567409437826426\ 714908252124146373802589915655209922713802626858161427167459139544814602\ 423020555559712935177174663672973389405888537206536519834524550681504561\ 659648050712152378805609690405013679683475452956415712884662531831530018\ 128877455529892218557552804844528098292560777110799445146091095455114532\ 546306942258834977457754884723965203098187671492703010378175388041261022\ 044615115263786855834715541340405236626236461399544470072448826088852250\ 045490558452234990519102309158148992558792431990836293455806797703212129\ 553733092757347215006152223482784830478953573173640983257410982400009021\ 291468167204913148147932468476384665610036391293180479660299759514966374\ 803895321978189703684376097903580423130458010258557435613252756025996897\ 052215531712181074914057336865311172485225582748516459714211465838469546\ 175267576821825782873964932025884116592568279476765105477245728748618427\ 961047842599587914701706089182800251556302229101029482442075032688842238\ 626718891307403279232206552886958488926644173562894178383643922522375852\ 938120469093192467161711404061690397036035061208575688174720780939271894\ 354082501971091779226680111655907458068036499488316225923819556347237165\ 362866384107172196459315081172778635383704958201054770417302001893046550\ 537614356512963509988656681526970314716146663174568342914943693927541364\ 875913258901272566647686754561694329624794431708861378423408184094953623\ 909326963147913594037509885655590531385867542652610276243079530303890407\ 683548220317510716947619657033006658340932326449688148757495690108463044\ 135223928846772866863598218093588786628313579221079640680203612926179428\ 251604878241242023354922634580736377837382263630508322105363024481278838\ 167020695803011919214742945308374309246234776209291406789054316711559031\ 711248709578303333176986001012089028391287057668976177565094126819539651\ 344546616055009467386610196970431099540544957522658768195608239841072296\ 271831726283379016920463186445651565829942722848519469744564863428096761\ 655804357334144083767068525967020655154029988050018957547330092131224919\ 256874167300340398845259023027690740433868916614698017888029419106421720\ 422067026459402586638966539050403338337439894286805858762890004441015230\ 653004199801605207562436651890486448358149171146048414452445179187019701\ 807269496683338707186895864023635412233775271898713982603553481399707595\ 431892686931355753399125873855506799707732644318544752907782847225564 -809 10 The smallest empirical delta from, 100, to , 200, is 0.1696637544 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 )| |-------------------| | 1/2| \-12168 + 1378 78 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |78 (10 + 4 6 )| |-------------------| | 1/2| \-12168 + 1378 78 / trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 13 1/2 n B1(n), is of the order , (13/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(13/2) That in floating-point is, 0.158647687 It follows that an irrationality measure for c is 1/2 2 ln(53 + 6 78 ) ------------------------------- 1/2 -2 ln(2) + ln(53 + 6 78 ) - 2 that equals, approximately 7.303275005 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 23, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 27 1/2 1/2 3 that happens to be equal to, 3 3 arctan(----), alias, 9 0.98792161444730336100 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(55 + 12 21 ) ---------------------, that equals, 3.4812632908955237249 1/2 ln(55 + 12 21 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 27 / 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) 2 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 2 27 (4 n - 3) (220 n - 330 n + 69) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 729 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1107 c - 2187/2 and in Maple format E(n) = 27*(4*n-3)*(220*n^2-330*n+69)/(4*n-5)/(2*n-1)/n*E(n-1)-729*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1107*c-2187/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 27 (4 n - 3) (220 n - 330 n + 69) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 729 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1107 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 27*(4*n-3)*(220*n^2-330*n+69)/(4*n-5)/(2*n-1)/n*B(n-1)-729*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1107 and 2 27 (4 n - 3) (220 n - 330 n + 69) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 729 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -2187/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 27*(4*n-3)*(220*n^2-330*n+69)/(4*n-5)/(2*n-1)/n*A(n-1)-729*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2187/2 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 1843560823397038997675743667264883218899438005757982724037925832031756244316\ 152366423351975228567562279324361070988423562281530639299769163031911985\ 746758511243918560383532113583996315666371316717633971149819387243948677\ 243543874370978535864182072124598927037135631611895183105802266362550629\ 484811925755942080990319206476403993175827530813123180198876003912035730\ 807097463692309891641527545148801142434555374366576482923435223004654495\ 750940338566763729908764528106765099736266230716337949820529467917672400\ / 99818957459871364766632268531162706341698388962496531140614820671881 / / 186610030232856719384155824585818003311216083544130963047420375155769898\ 012003361060756936194682568201031212630755073661604063179961700973641468\ 441123568305177308186115786820496863907681850796394660515008940973312916\ 288007915592170379930420985562364991658484413134495919999225824342170155\ 598514912743403633803353507126054407492900519640023399620733860127469447\ 514566724026222845880316679948529519892622540562388780858569751909947266\ 612561604893925903892162502514860910740933510890655057330149630153113557\ 140456260462321134638037111319106717252212721793771002701541564614688000 and its differene from c is 0.44583248497997926693437421579552381997711682068962825717523172633390847269\ 529896924490405669134355945153818435574776317190250026627580301224081407\ 805288875798363683334743452335730763735884192354865935201485893766791777\ 205186625652188959582731909639730517957776353163366670964964295060836178\ 414082491602019831222024673934309901563905296665816001386270439913469474\ 528955932166892819834794136582716185114200550134756349562279259145455515\ 106573119120309486002254995679643133913175015573956993611278196750808757\ 869857492514091190436357075086707147854881936361588657311769799942826017\ 166035522269445502744387150540082103343809346668330372605780959489963059\ 751579835803738678447273574198267741680878181424440167632693512000398557\ 453691459887198418657813053687115785877076191892931212610639806859608177\ 217173923314212768011168229334795068038914649576682438069518651771887326\ 065969638319883164909395753210861259465219880661005570234925046623958381\ 611752865176389045104755248424989655360323886751151920820443622478006673\ 441959149037412737551545621732651260126036058630349680572251125164831199\ 610504614899471519587816510067607314828390581601323265245120896656722896\ 749579223013227234114336177388063098752289748201377735477089150980601786\ 837793740408762862813443791312644892804589326118284254478927361956428097\ 827560018503746681201525860228893817517263007491657190532170921086256556\ 435036906272264837141714831706175798388419718682690168630648368527265794\ 969673609763425837151148621929119134488316439341570920150052799607172743\ 848641200366270696101910777183694504707093836073313056884607638497153327\ 502582036588366451952273536224719933117260905735611930727251008555085261\ 278138781347715584402532093026059336182984764569819866077508095913742953\ 437017015697370664292781172587747003437472996900192433256877588359149054\ 108758174257431877038940880783416386934524842588291444903230328005700725\ 287641858320855763459424803901652943315131796608812295129204095680702099\ 660769334324317117699492921322716207492802161060590551876075473646653866\ 518172816509847856317198526816480742706477926799166956742364442234634386\ 197598557792723343283651539358089564403982036406351086476186115361288499\ 965965996810241718174535452898846888173608650581700901948374567498791931\ 350440160136307253558145595889762419953750535789051870978325411251694532\ 428764811256103091601813205410521587956009324662162376477560109719173799\ 693416051087603339530350355916466976381060930178653169398119338018313739\ 107108974450739548819724586015707829468507124394671307115616396868846665\ 823678153771161064342111252396075949679307835183144364553443204150181830\ 046115874461877680318973933972767836305595283647618866868758375042196450\ 711103978184835344637387391353952989081547120757907493391811842837593692\ 292648673088642043177215051928567345724685356109125835601129297536356453\ 793999369745629669523258339107018952568906474777054658644871202616897043\ 884398781950920881654194407376229325823449471725873835268943306021726897\ 220867972157183726747074956809382523575653532906713125454302239247358281\ 449034359101667977254003135152993543281392109861646215025561227441743601\ 076912546324688671423660931856245165144848517018172863252740770499150413\ 930041672935660967480732073947011964502870668175486134907001059452247892\ 000800149972143196790740328831121621239542912610902849051463083905223423\ 856330488930466135848465968232769963574091354953596352165976331022885347\ 905096501176393607874421195622209788420155120923077335577052500419613039\ 474370314538465852754576225788669626489967017632794396091274157609315505\ 979840328439164977343132482283527459709219540407435498945490210044487330\ 283622480972683248046124861561491734548924570176221950527307299550439504\ 631072278325043700877096611474475191766879767854414876351847504991508552\ 716250673085489999834191691834679841138168166568001007906178965392944399\ 455684048585580749782636427204301992298563318060398168820274111417944286\ 791700620735619531169577079143804777016505622409018363215032992407356414\ 225495261336158929783043582808907620093464723931570883005435766338262517\ 032972104215319795333909818148424237676817674274143705547125673006931963\ 993940668108130447180059486218561595022455976716651485457420753331513745\ 231949171075646878286240129359657707789484527447642481351851098619712723\ 770829053855499572159641423351323225272495768560949440603502848125392586\ 998987576507587469678810178197084743543658615887344899643995427624588812\ 505684401741726630809748507554093725471499366660488982413479278320039893\ 324263675548605739019767986604210623798778379468251996904906107206104905\ 546101771173791774532157808819344444304043506650199010242155755059046356\ 021654269336829050131941809196971104887123244183619502186398536345156136\ 700350410515774128372390889024855755382462095042095679043248474265073074\ 592095324087687829827332144153223135784795040578859720213277542716339962\ 796230053335853902945565878280846818697580420451277917701781944198000864\ 282365500851829368512791841850478806863540051587847280913119305386277973\ 599728636370345456031731694142834274479503695105513850591375030981592512\ 677524338720302149380416556531274316013928883324759815098553510067398973\ 105023628278108371788882610723617054739787884529939666872201652124799451\ 576048726174814058299145516424152154673100848763721881164185752373162876\ 647763561220379568530910806959512383257311023675767189345937506428597229\ 275444996104592021176548089089598116933900149257667507717524539542199515\ 870310239767351170073296473511752561835207854753098522797784770038448243\ 497830193684889837016675049484741490080183113535357660306416138273054877\ 623777232874922646185467631326325944061444929174719068896901115292553662\ 773188760565279049181059581410052976494727664609774093155784630264827183\ 195542261629705562197693999257750331315062000153116162609561357936611950\ 449666514377310152232963485113337044255704773621183941138609041467186998\ 378827996263886622412737753935329384340518398338316072218274169520097331\ 880297435372603825418614150896959212175378932333138277928618842884136915\ 634797375022893920232516129994978832758462325282989857799969244216968283\ 607723095233305716552519249435247568835064011139910891750751598682363542\ -816 67057362803958297811202919536746439468242561360015676602610653 10 The smallest empirical delta from, 100, to , 200, is 0.4116553245 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2 | \-6804 + 1485 21 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |21 (10 + 4 6 )| |-------------------| | 1/2 | \-6804 + 1485 21 / B(n) d(2 n) But , B1(n) = -----------, hence n 27 1/2 n B1(n), is of the order , (27 exp(2) (10 + 4 6 )) 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 ) + 2 - 3 ln(3) That in floating-point is, 0.403020513 It follows that an irrationality measure for c is 1/2 2 ln(55 + 12 21 ) --------------------- 1/2 ln(55 + 12 21 ) - 2 that equals, approximately 3.481263280 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 24, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 28 1/2 1/2 7 that happens to be equal to, 2 7 arctan(----), alias, 14 0.98834400807312791320 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(57 + 4 203 ) ---------------------------------------, that equals, 7.0173748648955740588 1/2 -ln(28) + ln(57 + 4 203 ) - 2 + ln(7) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 28 / 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) 2 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 2 14 (4 n - 3) (456 n - 684 n + 143) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 784 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1190 c - 1176 and in Maple format E(n) = 14*(4*n-3)*(456*n^2-684*n+143)/(4*n-5)/(2*n-1)/n*E(n-1)-784*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1190*c-1176 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 14 (4 n - 3) (456 n - 684 n + 143) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 784 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1190 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 14*(4*n-3)*(456*n^2-684*n+143)/(4*n-5)/(2*n-1)/n*B(n-1)-784*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1190 and 2 14 (4 n - 3) (456 n - 684 n + 143) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 784 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1176 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 14*(4*n-3)*(456*n^2-684*n+143)/(4*n-5)/(2*n-1)/n*A(n-1)-784*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1176 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 8546244758506699176899676040373189772054169218985480198465043886061533959222\ 578052044569093207239836202849957933123618297985462732902088012697225134\ 250680935127988408557308337619745081991220185284457086924714018593372886\ 333529902556896004937864071685843281079817856845068203630304273609013883\ 229833632234356419789890823924694629853185271662365782613288646887003145\ 587533036724043030949505448329560910061317681079579558449767489263047675\ 853803561230377191208030112944801708100706010428823569286639523104055445\ 225041197616547998231937666629188795092383172429008845746014903351922329\ 432258835245371526990787644328179835041954120841042782281847392365933568\ / 042087774846270233126799122235423392302268 / 8647034523099329182916162\ / 189708649426181588849083224217427783009740978303572721860644077811354107\ 361172010082231524408247910113551022709153679254398125880973312390980444\ 409862224881769794931693824963455911451699552822080594752108384650310567\ 484097264782508644112879222978583145333749684557783672262654827812605624\ 974484813988219070338929797108296045549697116435924176788404300564687491\ 821036743071477382883502308078939932972145181414071505316496404780577076\ 957635377504657878690656572273402010209763364283494912530263612882603566\ 653347826844230304361524333331875126924934522362515631328225000742534387\ 144626581256524996810978410090980076106737290889869565454861368147086556\ 991741996060323865875 and its differene from c is 0.27762260304789837726740920108340772582676843540414894442353769412832324958\ 611895525656980196659326532237509204494972264653085670582648739254548900\ 432795483009316148590972921140700288138958804815005789028570950927865249\ 180332406026311675059881039469227948109296212708157299746422542261020617\ 522710223534727019425883704918163010618422958724027048078854088491443675\ 003880959400851797553565687352502493018967871819865633364769727663236911\ 638136898577745607927171513471909624489527660297654358193492616168729363\ 548380824877560465278804324377151581293828471201143805988134544645036696\ 579535513160105915790929846686045157406646707432087178748357064847548656\ 931176433611633388206826472675188818517945537928476831483601499484556382\ 929457626272488448387449586681454853465408216970128700557348800213508743\ 532897812348170046101745242862297744605696025236666454875033310530929541\ 545865971000619321694398430054605415473622097306387102485418790723693344\ 743103670091836605472660754928437376740910799628726083033444904779243421\ 496862894555381248728405239823379213692229172254209511844818931506489760\ 915200861974466242099867888227767019581653365725398867155421790122820774\ 020679835915457780330029336872483234575684425421183632711922360307690069\ 225166787735034044499611190280707506964863788439886387858469887864653689\ 441778816392727591184434974189160102293582496210079530056357679346354672\ 266004416676996649659348496029458750485154721161188868843219763809881785\ 520359272602878096890412595139669921776321608186419869450748882987718897\ 907597027588870871721127456148865482150010594753396401469821515544062039\ 271684481828908845688901185876251222515872681900955904417365410089639188\ 109167873554914429496352029882056337143308483042007460892700489312850180\ 434172468579781394798682926806176932166757916768682475642500596301426474\ 655295346935049960956535808801416266687202921715308722674847174827025400\ 044720641353583140372623905661141995568059793465076569366641310548226061\ 767817900808319201694794728226717271091342142086782785199695674286761791\ 398835596071371390924373595751310720481842028907652377486683933783176614\ 676924044426419604958166187509733597398376305064758446430648793622625069\ 334488276036569683612267738411664694774884578513189663116849315040238950\ 167422587008559312352988207640615715180606188243102384589103033352743876\ 984103651256687685690635966030591153983271952136042097257364918504539746\ 711550798723494322447944256625842609063496890140191278266010228615879126\ 664875076526619836815291984037487393158851418421978283254409767115681470\ 427504373945415784485916063105278142150719888236116439133348444041026541\ 587797098815119568883580378349774599913618898486054110879786989842628244\ 576678996390664577953363243701359777739019955018875062139953866412420686\ 570749006922676407836698234288983734078190198747429679322253154043673943\ 403849491425680575590419403693131108737315759110626793710351417362869503\ 277782747026216398706050498395083915571707872360122910015248471949885401\ 100815372367033899669251773538735172799383304839907319969589435422295247\ 650798061850574914212898550630719631795915001481218335664242127300111307\ 223720326581290445370717128437959992600416342789167927694117286303754239\ 383242813792847917360827858439306012305947967067549600357543140714885505\ 758015938059660566083536676993385534592894463868384761010860461937242735\ 793424052909158747245562758557771291745026570243172829787022769515994027\ 145303136960359713601902090126746832171452308537681266094093764637850699\ 620609999993209696513382838826372888857055201275715662390986037790608252\ 071526297545095777981117843666474204576956401023703578566167944423105720\ 149355887590180789376008141758655367989268376537746559722378458282876193\ 290614527350336724531498134165537236660324404541515092905431705618437035\ 409118247229355610930995764274819122393619614412135298555117436808888938\ 387683029774946053470052122126913076201053829266279386391318168813133046\ 313955705429551193529267179311446663463071795078414052059049705581628955\ 807497541434285773798707003064980605733468685395697104505617238122884248\ 670004011958846278283828111974934145462944839850313379236810067287092863\ 366705846064737661691767351169549618717746263829273923061493503135585506\ 091966028172922716930945664866979240609077728716772665185900317106793594\ 720820866432265138787593875878003714307035332078903774697658847737289571\ 211800397395707814284378768808936419281345045320093020929729056182751015\ 241114975380878528881391016754320812679671470902348571347859728751669433\ 720293520377144968530072139194547330413983142565767362614027970694124357\ 559743065223800281126647764108625544018142295425214789564543392454024901\ 444983119501640852238751714790080214518313730554066300877230522652705486\ 976092122892134998702671745757859953902388390537973365474354820401163421\ 050211543182600345807757593717800875425249900363554792125950630225725234\ 002294749552092312856193592539520263559255774125248212423941600476663379\ 079476666038299490495264759899767702923740641575351587425862086305415797\ 522287627011047856157449099653831459874921235688624253774880926210874978\ 930088898851849971673698198294289437382734766412332973053993848979083253\ 165400917919877820923905687919160777506699352452128361764799913156307953\ 065666729119257136613986108116127029409922365603872401607151093315836345\ 097577240983985219521956012052675349680124099779925693598852496099681285\ 024606832950425557763534801226324592104833342987567025909517945760166783\ 872409618125384788326434958064536045742444255789160915521985659839967098\ 796630125762262101254309233190191282236403335738384866402066973173077671\ 374821651664415205207592789225653836341396793268693583573031548558889086\ 523014882765872066660932838538126317083979508223594329128692796363366576\ 498117375511182826256430089951877175322647254242423731477059203764405259\ 784815090619586139983907865055252546327114654140274133242364208747836727\ 514627057263341392575411925713742275597793034022190375169634507360349144\ 302558003278884229599862305506713214474246233829918670509446158206927743\ 143180660746368465320196655628912694933093357467796255385815637638407751\ 095221881392070578117158221631043380544360110171457475882853606221608028\ -822 97940329079686783358310856121340219843976079984489825006 10 The smallest empirical delta from, 100, to , 200, is 0.1778061924 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |--------------------| | 1/2| \-22736 + 1596 203 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |203 (10 + 4 6 )| |--------------------| | 1/2| \-22736 + 1596 203 / B(n) d(2 n) But , B1(n) = -----------, hence n 7 1/2 n B1(n), is of the order , (7 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(7) That in floating-point is, 0.166185448 It follows that an irrationality measure for c is 1/2 2 ln(57 + 4 203 ) --------------------------------------- 1/2 -ln(28) + ln(57 + 4 203 ) - 2 + ln(7) that equals, approximately 7.017374036 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 25, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 29 1/2 1/2 29 that happens to be equal to, 29 arctan(-----), alias, 29 0.98873785460279402346 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(59 + 2 870 ) - ------------------------------, that equals, 4.5927237087173453540 1/2 -ln(59 + 2 870 ) + 2 + ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 29 / 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) 2 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 2 58 (4 n - 3) (118 n - 177 n + 37) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 841 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1276 c - 2523/2 and in Maple format E(n) = 58*(4*n-3)*(118*n^2-177*n+37)/(4*n-5)/(2*n-1)/n*E(n-1)-841*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1276*c-2523/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 58 (4 n - 3) (118 n - 177 n + 37) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 841 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1276 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 58*(4*n-3)*(118*n^2-177*n+37)/(4*n-5)/(2*n-1)/n*B(n-1)-841*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1276 and 2 58 (4 n - 3) (118 n - 177 n + 37) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 841 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -2523/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 58*(4*n-3)*(118*n^2-177*n+37)/(4*n-5)/(2*n-1)/n*A(n-1)-841*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2523/2 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 4437266308395781889827716646052335890924626615890813267888839697366676483035\ 694894459424181612593120481929301014719985966338360219904034460156165861\ 692767847270015871737107319330438267562169942784312845625551315778895408\ 959530596261394756940024886088291771918319244637216688531023048246606827\ 391019740622329214793388546456061485165157422845702164671099478968581594\ 749790918610913453658358778897328959216462213400334716856265279681001215\ 626156595777468227882603491710912737844621271026529368825756055668466893\ 551898485601189759085447501313774546060939222341426081168746624251690178\ / 4984984423076958882156292290613043647340934983732967924993 / 448780866\ / 206277326353948684312832637256875770815482579143731924963279207570711601\ 669800813186597966654928083432543910662112240637117963251748450132244859\ 986476973096488446039066504928245004121361644784190548610583472509967477\ 177924101326215635218733296153747893750216574096211339048955253431894224\ 206586626927835624250832784838823601135741617519189271960008031625487079\ 301385394574738709792288302224579548489327015610743748060909072708931277\ 818014701909210667264840504523817972773076763652320607038860010391665653\ 981640105155361501572740714290791209777580570165977633415063061687770156\ 27689062743327873650147451130142668007113849174429125 and its differene from c is 0.28302936011167949004191408866105087703265000141156577009554411597490184412\ 054495800519863341891079510539973216765144616101709484149113014477056835\ 127181820669474325904134411664401178730085361451699094631209880300497556\ 347626153792404785583072920022699856769241522431724487056250705013878673\ 304441516895341723257890951638521574616406611518561817547310095213250718\ 099214945912121355196029446895655755096346111293327829069927581434994080\ 562139848856745863825355984540703268156683368334131046352119193990468629\ 464735046042562718589506413800125720918417081112801063696279758433127799\ 226294961036233380628948511312613671071191613203765443095696173071917911\ 813140247033311231384145237922480691560820279172414405691828863336525186\ 258911894525384602518046502162626262391689909005815276193443913996527854\ 109978502694576628166359167324033779129270105498418101618852982980189273\ 728649010815316875956035826676536833405719433385583632814540129426359215\ 255468470350371684050506149193284764644400512388919966463359417947755993\ 792729681451356973970830985776372762137007262718652275937635709720448701\ 026549954752977823689236450782691692192068470821956331154213484994557882\ 981573807548478258274481029060588829114844520509687327969228681443681040\ 968281865328199460016999769499177371293913835433777915406030518178465035\ 801614609654921542332414293536917614037088707624470861806315571115826123\ 577972907006121113717200924955118206307689043784194209203313385438699035\ 471821127501797457388849521901043400798034912096897481312768296799615716\ 448358900903985835662721766053354265051505169650272286354636636836807351\ 027766280487161429122077467894644050844679492093621353183065349183672892\ 523075637215667163577235406678734313651896677254637950183762826516409363\ 174724480437245134905489068826279006843431786997461721773794914783864254\ 508683742931310692951489727100239304545103781583854196739433543220508841\ 599882980199809935494403284281289853990740355334190453256304696762822331\ 615122985890002597120052862730618541007952207615540876581611048229202031\ 488661409369358659362997326106041222620146740734825287475594621224160472\ 305001003462743891540027812833603410510102405842313813139666418728941221\ 457120665151068178240141695848831952010277217601563823250256527541568364\ 639260107475925123532843384111013152528795952234715939369899104776436332\ 480278996850908867650745358335269750411931480774228811740358172418518694\ 971726927482084812369111181335999580093799752535002206295862376378815035\ 183076210168820917116660091354898653143034773941006586876050218381709638\ 240248082515174168506953960613678232672039904142698960609553984103030142\ 255010247356857146041695995562768424164327935694184041998453571420545719\ 993690640066201005457283388603167650463031343855496071533329794119860548\ 033632432740873651977122367472328384210769306973945923018717801692842803\ 280274869275489465899010562419711621075569050635234815782893699218806137\ 744785969350788199614557116468100819845918001841417409080049031260596991\ 925497835562824331068026611830717199236501186531495608392424643558182184\ 228659855351673052180772065362263732815697633247175858580079753321847982\ 048719241812809003629264957191880127302564682377053589147926685323640038\ 360377919273069127394871537253739384898289752169985558227960173129965527\ 165433453286022503598212710871192804473609030597930693211901631024891583\ 174212949910825445481339486998573692926806015666651341816807832405488165\ 596993489987487839867280370897229044830762597195075129090304034153446316\ 155351512156831082935453144928495389602020306611557105512874462117627314\ 675351617118973946117266488284604431643793241050149944281121910483338145\ 177495218297903975248853662880298872238892625856530312106623016726783276\ 908578347025222471905393051868640064992176296328300219240292420877544465\ 709389937153725899268273686494844003740446506003842336819903367527561709\ 911845818998698094177419423034679545378502796595950838277077747128884480\ 547862122614046894376927538918714456605780602879259238128186548624868580\ 356561907575771601858202236082308484817549489927266117796716880803567819\ 330894693655185050558747069427636587204114877557263995902025859220651781\ 272901765172479522179535960305236894937366718296450859851409631053350804\ 224014314823849166965363118318407989096548684425260346497962654386708926\ 181901931671833701739131017532530180218805516533692421863374612915173895\ 353594524006290570019686244119419421967566949225236163385413312741492910\ 278701902642446520682460237085894303711042204107039204919416266551166490\ 938259388328822968156718119186098607567826971100862044936768012190223978\ 889703605446060763712396810561993826138203919925199671115609731080107950\ 529030158242180647879013078190047853095846758765304159696468611152851073\ 721680103408250095282273382994553452658667520584536118094910296341349513\ 361989504785845779957428061724198590575799680298074693188954433934247443\ 389522868754106882597393036557958305924432608822612481944281497292449866\ 362932285390164849943438242788534682379893951927400086046421341745924884\ 434714993465526205472578223450661660928550504728665577682164938852098998\ 850066458428436195361633233940830624813419788693741163850814091454029115\ 253045851999798388208173064785520744046226535637062318497597876971926775\ 236023022869893891707238535505182405986073728070435063235853532073237972\ 256093825314814509127083703778311374737996199632908688281446397455853234\ 689437084000813770743850205174455837356318249322074649260717842965673013\ 719216253681383161103170471463211560639737462698677419843791293988395423\ 281256926947914210615139518213605248250511466003038750265700423760228188\ 881247366862624643250379407010295492644668148578683019743094543665567763\ 172227014643800701482677803245128983817852019956340444249544083578118762\ 589275755043006308284085456081094718615444850362812539784624486627592666\ 583481851907183154057964124692606486653548803693205775863300294448578070\ 568039247240456754139857191120035138709640677195444434555883489712140910\ 056229501508505021551279589397749758974545417725732814193015263577126332\ 770619034639262217616400404623697508148903534710634634397133421835575847\ 812093585378350842974936932237856872014911589499592573066906518385370152\ -828 01914009540339094598737338059499073308833238930244 10 The smallest empirical delta from, 100, to , 200, is 0.2915728411 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 870 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 29 1/2 n B1(n), is of the order , (29/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(29/2) That in floating-point is, 0.278340357 It follows that an irrationality measure for c is 1/2 2 ln(59 + 2 870 ) - ------------------------------ 1/2 -ln(59 + 2 870 ) + 2 + ln(2) that equals, approximately 4.592723710 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 26, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 30 1/2 1/2 30 that happens to be equal to, 30 arctan(-----), alias, 30 0.98910595364147387077 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(61 + 2 930 ) - -------------------------------, that equals, 6.7773028482094062256 1/2 2 ln(2) - ln(61 + 2 930 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 30 / 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) 2 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 2 15 (4 n - 3) (488 n - 732 n + 153) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 900 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1365 c - 1350 and in Maple format E(n) = 15*(4*n-3)*(488*n^2-732*n+153)/(4*n-5)/(2*n-1)/n*E(n-1)-900*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1365*c-1350 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 15 (4 n - 3) (488 n - 732 n + 153) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 900 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1365 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 15*(4*n-3)*(488*n^2-732*n+153)/(4*n-5)/(2*n-1)/n*B(n-1)-900*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1365 and 2 15 (4 n - 3) (488 n - 732 n + 153) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 900 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1350 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 15*(4*n-3)*(488*n^2-732*n+153)/(4*n-5)/(2*n-1)/n*A(n-1)-900*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1350 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 1747962851109595579341884687356359441232573378996456458465783733654932858013\ 776416271795744246738082232875028859206133270182750210529664339043670620\ 253420173762076170644179420926617841401729650945699204869389880805517545\ 934152370016456226118835324086050450370234389320016575841462895962873123\ 953975047521392780459401122615896949278003740123923951426915079729159411\ 367842755174993704932298874673914516637027918972994829996937887352612599\ 777273295017373739962484680604398767882519202250204262381626761601629475\ 320063402702065691426117440343796235557060982254548463966353638518647569\ 492054259145038086887480119079266569871783592092299676591191572868158298\ / 93659447516346172362193994361652963151018762 / 17672149729503987225321\ / 238730576815367708634547929024090682009486957250305631095302263634408686\ 188911488992693677001991923399057909707398234944927931308004313029009636\ 032982118076370209685059012643361237788539835762283217393207443165224721\ 331610726038580773673380217805904067678899238763148364501928911115144651\ 082167250210735286746844501331254482326856752281998707421862130635310869\ 727380863983311604983650370739793588120070081468205598910066716779484724\ 232378600384402616549642284228853152143936386213338887101305226020198382\ 099662190371695300672107559244044148865287157122806083320896701802075597\ 648933768226863093411893403115734404106078138067478250930714982603405521\ 2967612880049498172722935 and its differene from c is 0.45710801951834759174434919009172228867703777186412117588729593913548187201\ 410801975857960481087422145850707529382900760677568111374433276115065187\ 414502479146876726727380319202541197752952312746834785365463276993160354\ 946853857989912632128876569150662052045142613844983521957814549737274992\ 430792721654109551531102245502574347569318801656189203301872279732765117\ 092811510963324997340742438413403075417415204137310398768496976454076871\ 579168659292588032697417392593492247501258261737003692447454706384074097\ 014067048356814352499658715012224108606053761159427866745272073692803858\ 437702581845625418257905677119554678097062610526297783411415457406081783\ 465011106599345143141691622196792313880589590283588418138508793493606936\ 477609329068065453265398743880219653528664193120657300846179173007100002\ 863007832333366381938897690816717929186351326194048697808785257625225556\ 846768882913659132123982327280554590549931999530482759958334537184580765\ 419728036879035614290197897120398425896357635150986672363686022145404288\ 002172955805513716758882103704437542259233770282366136051046463304064696\ 770950459326327305281465656473926437319538417354720123894318040352199799\ 187343881142498444966683939420608156897838710775412756791038398452354667\ 595480272115841668688355143372298725688015767190084061477707866667239886\ 302967665163974967849790067574321787092860017968905110308326253422362997\ 755713113714557525487973862345232809236237244968492174482583894167299767\ 519332861069445531487603251190968807863962735098155582115893167496805178\ 017964780177456854131550805944532567298860485126009502599419094235995078\ 971651398462315591108814786275895450004708169909185663988816347081319397\ 423951810646873719092246048990406032887865886884975875063882360350196985\ 235614952091370068626105399647381676734470408242844085981582844814542933\ 428727341723812866863424634970704245641550870481950655371116795264189756\ 930202970874526519895950124618878201792188947128028081540647170456050602\ 969633436195629193011100307083423767960005790551962748631987369741036659\ 202686943161501126459519137045891129386603412815602217662912502423992076\ 918734336643377857487480212252072586315857127869601951283111774276712578\ 764101053067810496352825910520465766817701145107566048254953487501182149\ 759827944370093163922162051332594867255435814496051261344432499595742521\ 197415608156152705815425151190416840062194907325792533922782783910360037\ 567226894510129252588339107633225503816826583535643438057975291034715560\ 330859262353396047687560473330472110403472814989395217619379496307991199\ 495657940624020825067656262705408514488019628358388366180295101856515241\ 126506951299668713338367037655265520072073080640511559116983821428343071\ 413207773978520124863217815251664672625343355632955229879493349611279748\ 037878531639352838873788465293561482934192937455103783869055472016772014\ 653302181548945505426676993007950524074344054739825011311057310425179535\ 870114680809732592594661252241111866989358615459558604126632640576528706\ 768677670763781100331360668442263675946126456826764784459990656896011225\ 258029449185236947202154480073655993660869912131848501186289992480851232\ 216647114251972616348617212417914563901966779568659513619231221874847405\ 201907536249617130656214004824046818949741193786732392636553256091099901\ 866510387091179991324012536145516880149278157231692801352812919194347997\ 819023901040426450544288109377061299421700678681620263163328717060819869\ 190164866019529896904462426106407104106157250231321709518609712309011712\ 309479165328967025926800572698638518846038310330588504992160281588329767\ 023302066827172516862462784342821862934411437793944622841491012336707003\ 749586287174368016504892208737988358979102784125851746424987535236408872\ 177758208117953019478100409464850364496809968894452529992580713262570326\ 481370830471903287091533212559140311871025905866830634261685104375015559\ 778257940675177544220793340218462159310851753771672812374499996594699965\ 665675795390085733816128131093128940757014686631207199678378981455818689\ 632742066471718766109304648285338249512392696682808932673932857885504498\ 005932000733719454337811846193307622145096825193900262027568050623710353\ 938550860134292282545642485585116888599822642939769967780935599684846356\ 423958886722054451935222731258055980060364855146534271066993729799319804\ 393954267365547244077083442831539639796665416678503444531347197811867062\ 664594625477402002127733427289342742056864443340994820424728026889582142\ 440398767608634496771302408496373748350137469916180594646344448790727035\ 087341278447803937182129621584185701627563648110278139499717453183269773\ 985843102088040935885904015079438472349286563212056924187935208159934159\ 327867810234643930431500155711528817002168586911223672965088127936997761\ 921748271496747975499612651427418022051318917563346213765509534853438763\ 537324385515477922229761235342603343619990669670525163645249578541214256\ 362382423632011991484915196644365986141449860781220439685004910105824190\ 968381294287850825207602690529695835865663061267070383086893002668833855\ 117408658798537606929999512404096943654236355818581295000588398393407004\ 693979073004263892162473673324614745390010397987806381916138111716368159\ 874850730540204844437153233715089014883702493489933856438602436402605398\ 520138890835778638182058997002452260325495134963142393025857074882678222\ 043593333714462716769899640428370468430448678848068039699511951393326171\ 440648266035068420300872698953631965293101347135501773896733967777365970\ 459251827210736132315013996911357766294177002469090448020835744944636999\ 793044712334278495937356467096787682814146139521413134437715368832763607\ 879815144378648979036573166928973463760126883862673622105294545608327323\ 135955208406641116645591425907188797397830385832373904226664425722461827\ 050655933254483194786941580977404109244451376913645493905489914792755736\ 039030028642277238208464402731396012524485097875628068340470917497652954\ 510311117324303023025311453455876843100615722554608824555926928850717259\ 348305028448247680981323842620335723812023384879218836198480282328203262\ 990850095624022950971756902632130845064416521084593173867082690162142155\ 364665700038133645878992950379377039185696352386163620171712750270401523\ -834 95749392844225017618239630046234946159908023 10 The smallest empirical delta from, 100, to , 200, is 0.1864049527 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 930 (10 + 4 6 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 15 1/2 n B1(n), is of the order , (15/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(15/2) That in floating-point is, 0.173091152 It follows that an irrationality measure for c is 1/2 2 ln(61 + 2 930 ) - ------------------------------- 1/2 2 ln(2) - ln(61 + 2 930 ) + 2 that equals, approximately 6.777302817 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 27, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 31 1/2 1/2 31 that happens to be equal to, 31 arctan(-----), alias, 31 0.98945075028252395784 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(63 + 8 62 ) --------------------, that equals, 3.4103283713008724946 1/2 ln(63 + 8 62 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 31 / 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) 2 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 2 31 (4 n - 3) (252 n - 378 n + 79) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 961 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1457 c - 2883/2 and in Maple format E(n) = 31*(4*n-3)*(252*n^2-378*n+79)/(4*n-5)/(2*n-1)/n*E(n-1)-961*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1457*c-2883/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 31 (4 n - 3) (252 n - 378 n + 79) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 961 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1457 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 31*(4*n-3)*(252*n^2-378*n+79)/(4*n-5)/(2*n-1)/n*B(n-1)-961*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1457 and 2 31 (4 n - 3) (252 n - 378 n + 79) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 961 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ---------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -2883/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 31*(4*n-3)*(252*n^2-378*n+79)/(4*n-5)/(2*n-1)/n*A(n-1)-961*(4*n-1)*(2*n-\ 3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2883/2 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 1800465058472762792571580873099065744500181025589272744391731029505731088756\ 953727482307756018522695511151509402257141574091644712285021967991355127\ 978560394607344976226317271640234375883202949778082236687780849619041240\ 848673355444205154710368569834766491920384986929146813077485307783798140\ 131578799409133031928363213853082025731828067159242676914720148365823079\ 594180450267203702145808328829616885182578054972804468254804799626030974\ 895903313704526869315679234862953261213615599980622277180241627755995087\ 720756251686924926806824217458332465444066191217954085729789536459959165\ / 389649 / 1819661118007808808665396247673293083574883808885158510785894\ / 104356128202149126048614843493853467869747174513610544506449718884454427\ 110836383508237732106008414146380241316361901524068861164118645295084238\ 412561900806531460930512358199293499824686670426661356172347810802802048\ 032242131550791422885165308335796551568146021542088377078077885848485053\ 744117029340113716643049645536500540534475115235133702186360681991773736\ 692070791713785500197356399957344302432319211845824728198763620957702974\ 536731773961702449877975343053339760612737716022565011100389492447380220\ 406066128543723808000 and its differene from c is 0.11353178111367659761184327749885002728118701711288189400846577370425506746\ 093076141020959587670980791055600781027024752260278152098036493987239604\ 279947366096717195843711455821840617477298412013572183423867849560204913\ 532006693003531774529569656954554766106531354809165307191611460916258307\ 691702270527016363639080206846551387861588626098021663871403182542448754\ 923696523691982134904591878071921571899362211136147711674800310659595546\ 135556751858992036649988437396634009483120399895737824179997604509121850\ 874655966987319104073514734819711597118387007484550071917533240417389965\ 434727549569280242898539826313079675356074531765555980933294598492456126\ 833719304914790499695837443611873471431759207973249702859154555779712506\ 659554437969664053958611448224589695379782165909229432468258500547121541\ 598567385586340048963702742933424425172220032447726727033286467042304951\ 866767469326697912779420597595404923464542359869542153120133441575733781\ 909526044633754026135672688581571254274724812909084418354863305292173225\ 084570549893730509393663112936725220979944082049184882290302071149537841\ 765936510877285885459706475193686294529275007103827381479704266995456130\ 386623524689321445425454192530722412846439396218642401334452650538124511\ 748059541259509236466576479932465003923136671049972659164500623894944822\ 737329755013155938622546819993446807858072766839758917290519540682485813\ 629596029776186386421996915589230160649440282165314379232429646296053717\ 449840562436396182088717243335240895910900985780309237048736409162543436\ 269155527132297654739214764849169838896547239708875788576213769082432693\ 671080797089254121821314969008201808396504226692048656435335924947342705\ 390648158397179725973206417826047458422608301304395137533930613661824399\ 139163656311676837967118071920287922535321206759068130495522145562558226\ 971144598251465240556816553694064341525617474333571416995391511538025258\ 567314018948275160553164640453788393578520380208084559540284148353148665\ 506850960013403648020525174170874962137942900244843051172621333885087101\ 731677569646804875317570585309424212677061770311380681182827873266864036\ 301173365201229937895409400476944553302387978644722599144404372534951134\ 577019266523009943404099553392634250307880459918785904151640434931534271\ 277151605940579430861029132281137943335013331584295182112576709795139412\ 154830234291113698065305223987202052440366940615088524339925625120946972\ 299841433057391115252420971331825691782194095517440063650430462114479347\ 329793432081860438161871960455245203493941135370233731126651562549961553\ 190553827771995790537837513285566962096634142155735413206905170238844904\ 907889279852866164583392955247806188231652909466197453949086484429501003\ 061602899294272146307134087896314372083378359982550968392717568380910807\ 198423705748608735007221026987520445558948216600951569158383245923894030\ 660787775428407436649402012896192292537920085730993156802460227825358436\ 140081507094836182057731300831185988993568288926713576139099881213858961\ 642509783188561513865798036827978581074620550246297173474823285881014091\ 894447859208160530050166614271826282230528426194932988023058105019864141\ 060013087710552447031150805061157472763323432327318296961394569166579091\ 150167586861458948430713213181423468179437266225458724538565176958041898\ 974937013537340435471855258715768699765298449127673545882612665274984255\ 976176008211250827786120281902932820493696919674178612165626428474461790\ 300337429179548006902594029402359109550137001889773673406591748147566978\ 928578625566282669341048354412713087642851368315052396658730699198110658\ 433095619188067111509597232881218547932600793636007569216859158097568713\ 121992087865915064337798385940387899802042456334782429503642700576932935\ 607338397893228803827793775218775584897258496874814822861138811922691214\ 186919361355806985969503829481578297179460571698279013237695144542793700\ 953871449205106039071431915389344582874791221313019921972005218466184084\ 910167935879529327866739223193279633278127057778311926258808704889271602\ 350122642907518158262739526182267594793965174578587925794304018755463147\ 669027539631617122735125172705693028819518731092153717396576930964384710\ 331701247205045520229493140367082187476896972511992094712393667507440634\ 255941624168053112992467583621042455079208524590904007774659118577969356\ 791026209750330808751952792378176294480542892879146410696238542211147500\ 175319023936538711556747120302614970293857442331028862153108818200986854\ 990412667166143902164049111658762325501413112021030164040304110935290994\ 827417584986298285228415168194803496895683752453871115583161165774877816\ 112412105060691753172591659400633613712844485007511681067850254288585170\ 975960839211336953691234636205060011282753755898376146858956425769199108\ 207280851421717879693368440442229569456003007276327409857177237549922505\ 928606099649032053994521268739640713493872037285720286832327903793705005\ 562591463787324248651931456845171251337077646468755721966404769278506051\ 238416569737966247510741792967387073105522716883165213211314675980594027\ 190899606558223811796785580977819126587109042464832110217391125508465549\ 054881245780459318416154267182174636162241238736326363972493626801985192\ 100057634730089519698125862566445902896079224565038613676042103362761592\ 031189045232036381733911839051037947988563091064418891236646716502051133\ 976645626853747104194133949408603129892557820345033951610082309769203197\ 129151729744166408921528911648658035454675450815317425056935976122756004\ 872895509149672927481924444158946141835669305758009289732286510529338864\ 134805003094358197197812332017276429853924423891793831959054784621000092\ 996172487381799792317387702410825635011153976910714833633092257045177853\ 185833559595276477831157575864410304090890723902855574828705947164033140\ 499623928365877442242788882989952857773716615431179278882529731008958457\ 953843870536692034777398858496931398338362250516536700648718917532335627\ 499391492487392828497543352602494394417532229705336802407235312856493654\ 572448484995465001832292858360195956821226262785913926493933661024015176\ 226076822748219398369954339699982280341367644753748734387268229756209851\ 608520457576900431734205214240468888872518953460315900139088974389650089\ -839 792843297976416652596928908085417674341 10 The smallest empirical delta from, 100, to , 200, is 0.4205293423 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2| \-15376 + 1953 62 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |62 (10 + 4 6 )| |-------------------| | 1/2| \-15376 + 1953 62 / B(n) d(2 n) But , B1(n) = -----------, hence n 31 1/2 n B1(n), is of the order , (31 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(31) That in floating-point is, 0.414881074 It follows that an irrationality measure for c is 1/2 2 ln(63 + 8 62 ) -------------------- 1/2 ln(63 + 8 62 ) - 2 that equals, approximately 3.410329279 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 28, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 32 1/2 1/2 2 that happens to be equal to, 4 2 arctan(----), alias, 8 0.98977438950357836260 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(65 + 8 66 ) - ------------------------------, that equals, 6.5724250597283510667 1/2 2 ln(2) - ln(65 + 8 66 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 32 / 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) 2 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 2 16 (4 n - 3) (520 n - 780 n + 163) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1024 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1552 c - 1536 and in Maple format E(n) = 16*(4*n-3)*(520*n^2-780*n+163)/(4*n-5)/(2*n-1)/n*E(n-1)-1024*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1552*c-1536 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 16 (4 n - 3) (520 n - 780 n + 163) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1024 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1552 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 16*(4*n-3)*(520*n^2-780*n+163)/(4*n-5)/(2*n-1)/n*B(n-1)-1024*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1552 and 2 16 (4 n - 3) (520 n - 780 n + 163) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1024 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1536 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 16*(4*n-3)*(520*n^2-780*n+163)/(4*n-5)/(2*n-1)/n*A(n-1)-1024*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1536 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 4211634726823943730838918662314109108706669384367533673727281849484942628207\ 697052778447324309823621407941184834714277434614187824362327696638712338\ 845820448534633545062040585065362577133697836953171882020022951630118406\ 814644496790441775860190017619737982863287579248205883349108389404114293\ 352442618154799392303476564916149227289716255380285139756535608039564065\ 266813565210672727910475932897037568526087894138078014501835605990509713\ 340968676774179075609354392457495036900029617549220496177712873811584405\ 885826630497233084690877978621931578607591229513894177090500784678702328\ 754603858177693106462840446570284344035095057512939012756087110683626793\ / 5210406455294929035338340515146416899428031686300177248 / 425514619441\ / 334542117414247575229161988458636969023954980355888199595049423260547873\ 815049649656040399704136788594970436645493781308374378778356100059581832\ 419639497514378553324971052344568323457567167941564592996416296839469197\ 905099124600347075637541619492595825888271788296563298041916307042317061\ 520440585624420768477721427786250604602616174314866661493724029133040017\ 604443730594886247025960630007186903019447500060735027769759879557384761\ 467389554653896453979052120034966894211926949926777907082484354023505955\ 324450417983712852951880233535986104672163851302836232268459954648356573\ 435865101317614410634074583060909801361241062978255402610962317652180200\ 41564773817847728748905642039654984177940077125 and its differene from c is 0.42212223495589058108660842303702934510216688931508127322722852341149789721\ 237495830616869407911301355212744357898410207473825394549378896317784294\ 695331454438303750026109826783919866355784677784857991976027891594269993\ 037704194756675057440579814278994795828428449660699945927483097922120285\ 600948648223005745554366338601041493512882589817979919471780908168616767\ 186456509512494936769246014367375062725570861723545250370201925997862472\ 753921857959386323299701930107192689824501004428027150255207528686480850\ 306483170404569416057780972128506607599036062115340770896314186732240984\ 910128288359951882139290419441350907555441753433165719045934456526652447\ 856087377311308948881662327426996395582552258293983400930866145302489427\ 714969263182112130463298956301967120325815574735092763219445753911255481\ 924073787380116527245748674180094296223916751043694398786453176086799313\ 638872126361324119352621868306982722056794629650490751862207963726284269\ 161423262039933788393922166775499640943560633955770404551616849889810597\ 469361136564014307035968556387371194777381847623022784645016338574280408\ 602774169592468419344125301391052671588028130133914876442792466128184086\ 642032351461720186099149505080787187145496227655978453601197595440783370\ 361170143458144900050480696128213055459962988283950359676094084721539454\ 656756915319512093265926506815757761439038101583345531720686329128394028\ 326707593506292013582007191575491784281421881278711052272678596853679744\ 006405782697030546415588526139934288707774170994583736956426109315000110\ 794917997225756308320190048014260018037983779510724599008965731003178398\ 762762614518296486368214959910136691521656423824300775345006371136833050\ 636667952366525727001831285206469863179569678726280466199998297527150673\ 623570501263201330866373156700198817148045707072589926784301670667846767\ 537774213603397599693745186733394852058117507742984855892519824069520711\ 729325072464267065628942564295189086659264073865593835700498958285990335\ 454144294261593838559829220439011260574297158832317832392466555489333341\ 476347352625728570578905767533537715220348213757257365571488346446025888\ 102298420047099625061189846572287416534359281347131902173928381157734626\ 203691882134856109710208766590277552734343256565593446894671805294396674\ 956526074580426951947474330619851161918479724083052551928836669351111963\ 131920721015423944470912944455421473555591167414798558859818953400201321\ 020106812457888418446042516904040969355436676116699584598668459930146164\ 341197392902830694317344002195161185848786530622662356021682164855187365\ 908801340231299182370746520828823954942831439498151493028238139717521290\ 594542490753939045657381631601178764630297181940220467909056607321125997\ 318480196973579716749129130305669617393446963201848213993036236580161975\ 216635523489712759794919266213217393325774959824030955726320579200237191\ 085321404994727527486180975494620004443124315516222894119284078899587219\ 061717860323001850450493454398806579226062548013563991210747704482584032\ 266368166937665710751412964417102475814764641677717036337175229182017576\ 364495688646005317620689016883643069010297689361746422436834167570297075\ 089572632831906725788710665272035497703334325935182194056421731596319858\ 237796813185431543314260282912865957339082822249197701850410693316059930\ 937348890868255028719659880994823273960707416225428022441737514847542121\ 859872503689379217655365154405703121197910083055949756201368006989781193\ 244611891003319673993991488109406170562545877958273390527108922924225206\ 624252329440068321725798115590413779515151856707752550512289973672810760\ 839485189576783155416680581699255044419995737571391412414459717598404513\ 080683703701431521654468301502749925653332475025321842597880789199843695\ 260682272559428685104617257417774983041882168363517774573235396209029833\ 168767744537569291149521957727929458459209865849680319627452195895183771\ 457286242008290151159148815922769251350280784844127239558189978720171877\ 756286268411213729369876327608068784922350835269410222518501595554297340\ 834618873147357388212662329838984128922010272667719731147538574317440109\ 764413222955746394323765658466309563247129958586675309727917329220394083\ 640173623692319346384325514876136434845077670723221831488301769400513329\ 428631747983526561268989392514072801080515338269945523345887318651548827\ 836041376649901507842815871982270627838438162834915488143831818356873796\ 295884384673651787014525017191472941334661200223232114213868410792957988\ 238637721093432324922887407468346521288976446717947234263674247432555895\ 147839237472574578972055161895521584456095605651649785936058463701448283\ 961183674931748354253545365307897627255014658727076562653488814243720452\ 974551076477648157110374336117184381162640533080789400716785698323217743\ 020409505651983474827280335936484290096876960665112305670488023943341425\ 019734413203371121105849990428803796354301268958542581957599095466059801\ 469619184475879796886714535076074340204599057541459783167328625821102352\ 822905018301412326741833719858084251819936204086879872318661171420234670\ 873719152402686300252349421577970896209646254247385427648906648084342812\ 481653877778463464423218757587483990604243673128713403612614645840050780\ 729503837663350672760909106546151504526590701554359306439686843197911699\ 973164140319101140874084227223105164190040154516053998440707610213647890\ 036477562143731169735330462621993719242925175796419662561972695574248484\ 222678602770098381418014557388975683720999974115679310659601513905365783\ 720736179544690777933554025635210427475211274395188852749519413351759216\ 771647617667154128077832954793440338733447700379023614976662866455308690\ 465089233751024439751631646375660940375802362319490498501808149966590736\ 025576891246406880215472635032180265035706967394924337143928924213949347\ 158946470079230621189212517573275335803864491663287546562966090216309483\ 387737726261706880251496855926595229515932965041676258962948101997640564\ 190557219961624842377990688265565647003931175718597142464805612306967524\ 051518107319446065861722080810639712873559225252329742312902173785995350\ 564808540556863782732782941993979613063840238514195789691271945642092072\ 915445797835825967660246905653883235998357554445083941245589393273057015\ -845 691696418739266691126719812304723 10 The smallest empirical delta from, 100, to , 200, is 0.1894012337 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |-------------------| | 1/2| \-16896 + 2080 66 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |66 (10 + 4 6 )| |-------------------| | 1/2| \-16896 + 2080 66 / B(n) d(2 n) But , B1(n) = -----------, hence n 8 1/2 n B1(n), is of the order , (8 exp(2) (10 + 4 6 )) 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 ) + 2 - 3 ln(2) That in floating-point is, 0.179455182 It follows that an irrationality measure for c is 1/2 2 ln(65 + 8 66 ) - ------------------------------ 1/2 2 ln(2) - ln(65 + 8 66 ) + 2 that equals, approximately 6.572421977 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 29, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 33 1/2 1/2 33 that happens to be equal to, 33 arctan(-----), alias, 33 0.99007876084444538661 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(67 + 2 1122 ) - -------------------------------, that equals, 4.4431661761678959592 1/2 -ln(67 + 2 1122 ) + 2 + ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 33 / 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) 2 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 2 66 (4 n - 3) (134 n - 201 n + 42) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1089 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1650 c - 3267/2 and in Maple format E(n) = 66*(4*n-3)*(134*n^2-201*n+42)/(4*n-5)/(2*n-1)/n*E(n-1)-1089*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1650*c-3267/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 66 (4 n - 3) (134 n - 201 n + 42) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1089 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1650 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 66*(4*n-3)*(134*n^2-201*n+42)/(4*n-5)/(2*n-1)/n*B(n-1)-1089*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1650 and 2 66 (4 n - 3) (134 n - 201 n + 42) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1089 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -3267/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 66*(4*n-3)*(134*n^2-201*n+42)/(4*n-5)/(2*n-1)/n*A(n-1)-1089*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3267/2 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 3463971273752421104893690942543910579182499583844517090105180510879582370549\ 417382568342819576067693864569880800313270762130421503351270914672630631\ 325578125536243081618300864797187665969816723756851112260300894108783146\ 199224870130754154818670621262356324461253758873793405794820085154494299\ 571912552346880248931750463705599327673918774892566638224237587526424865\ 908353171268822588633248773293612255341323090459084020866539964896455433\ 961621893349856332759452959025804173791944887308071803416937804233941840\ 058899320724329123660823850586039778242877802343505860586954627908196478\ / 5406087161666238123970661526874994433854517243595339546994395246363576 / / 349868253996073481548706574220746685075532880171800598394361903377166847\ 284239788533439201351063950420796817248422567027676166276936383698598774\ 980393760547936809000603913121542326728875386019709936352814823690182939\ 967114958019931868845985003943478503659998608171857465771862842796967620\ 467562143966229340852115409683456859468729840986739804660425788984574391\ 539705748117371470534457731684199356290703824728091965662365597165610205\ 900857682368824715017690716849301273476856581271875476035263314210675658\ 252164767148863490016119066158668485127326070824256888280414042762209334\ 157747639785025947707805931110168087671320946563749725575306325795651836\ 25 and its differene from c is 0.22927439656811152732759786883813464287804917912415588903800775802450554252\ 580069982545555852711375283121134327368086041644623497314113879476185689\ 404447222640287768102539897489259945790903508813779573005521680848008479\ 457889492033463881887732156825178888664218482896786925743671612011248003\ 224097982174915280289210846189871208248542069194327122149040560448345109\ 303853615214270581482610020944271641982049240109288037560119117730619549\ 882024999304614258093485383239598608294210904820378319240774048613867460\ 831314324783840723770495601882040482688982314221102032243538027521196010\ 113067549348230660865563590252992450653996440377205255470321077486340505\ 314579201004192251038375577053295484213634255462177730764255778407515897\ 209975418610838441440213018123874845365598576821560632005575623198679167\ 207154046164567288402029245535795412775725339042092315519985089729657242\ 729989115527854172018878389010944683230760406209191758953172767162869269\ 498582970213534973132468082535399713133369073765729300999734057502382398\ 223949830255094955324819236665628550161076536096463589947500214086711523\ 644002351027581162009361349341824824840668337428524190168543265882008178\ 820938295275904474455431356880168826465358746546466864248718362668712570\ 438018807631930947855105786865705402335324665732157542389661016629561783\ 150409924833167434702000511092185625640136951339988546196697911428804304\ 789050102294064281191635076932661952858263058119662819716960858585689644\ 313372235090208652837440549229987177029772872334644642996826039160970766\ 916744871295191975737429908671870944966301389405596172253531255783632477\ 902830461912127649898341743367946378228006245723068972760101296419125950\ 206124033475004775620216860884635476761871007083152628830891044871668410\ 419263523020448929613896524711617893416970389561345714073609542538934263\ 348760663004701026317988563593051261986287938772837891529388024077851957\ 218203750473795749813153360738662122640070716204112836124602815307104311\ 951792369908229470958529342118709883214838866656811130448802009581452779\ 346707267313205085431065453027108574632537023492616116564045402355648900\ 030096600338198542384448994327801556180581702445704519317639788038401845\ 773351713799253716608544561936298642360704393322685082909240233775659872\ 739493597852019013047194734285241023462900293987479293391774074646913478\ 954925060321597524777040101480937287541495784395180051495457608120626575\ 291497278799675153426064913141626142474577084170806769457400768951607017\ 614137480104191886479703658193318755452551044955754683356687163545811621\ 881119339750695422159725208792214469592023256347637894688077785551964541\ 984140405779277538604868779264631173516497288753596859548448271033014098\ 233471499220964788509609763724907539987091935370698640898016855609456314\ 136705976555419988425373838506955182209594221173276387692669623188359721\ 612377426878791877156835986270427241525866292403833350222283832384657464\ 059470568235272414667970167855448530433882281188398738508528107908389164\ 382171755885272671458071374497280079881174718599077147053160889614428962\ 994787377341564866787613821610880389876596563049238459121570499286120552\ 924832557631124222755378664371076943790199007222403856868423488029535908\ 709543189074860883578851963389477053816755524931008034735587064608192325\ 875056075377117654752755036513539636088104343873877377190997192460403093\ 777733012834962856700911245453155424380880123552961884347598553670919179\ 763846188958565295147632565755073305250704549614363883900905580967011238\ 662289175820708735905899053398300599363877521699127518451814874911496304\ 230837921595272233421803666460399482592421149680753292991898120436777736\ 449783082070212786292011989076771487051967189684266517237099944221003572\ 734488226672098186166939363302308523596541863581513647174557454127322069\ 082401878332965954639097704273614034256818176469504350808828792980747671\ 272970856724906247560862437604263297302977666131950012737834912146344829\ 542391040525880745330091545068454427293422896786897597187606828225552789\ 720015376087017992462857126181956380423185446708690998587537965533952078\ 015755839064986824280424106097498650352948384648694312471244629065407551\ 600122680814198223234240472045483506336843696894203743276411149619859566\ 812364752436716150385702265686999379502692679469009773049864618576643219\ 452510310740943601196692797541891494778606780981477359877773428335519316\ 182472468415255634487690532726048576076008816141285412078457524832800887\ 283280838943630523317751390862700416796700776058741108671274930813558517\ 298360388654754398616177112976124894394953826821426544134015978529174146\ 797599169247206970466111863251420440541098126447464522429269462177094915\ 390749209570709798838168973683824494375167854862196159875909815025547194\ 196915623881252220735301553293999392185653477927200509144449425350191617\ 424599927001317430444232309320367572200447244089302485798507951736129216\ 316996462426036356832790839245242917150278701750778778125934504093360786\ 044065769320865530891791081106100201067940681771683348487599847965710891\ 439418170461524468157548566488586000171813983100237478526182639795212394\ 194419298767813523177199921469376618545351802422584014002799244059098414\ 289156757884347049232756876970579588090501259727444923164365648924764351\ 627182202420781697198063339722450488056630286844938311303255059224641655\ 950549101062922472167191930782300901908305312523811585663562208704508741\ 672188825343109383713674621425732285937682811039155868311012991294965106\ 148529554813714667129694489751978379634090017846494521610356869071488188\ 152836378977874247218881114044492535915776851995019052337221333911917263\ 991092237306736625307355306025407590605453381525343384386016835107613044\ 622606313778162168743800145567337099487492633362618393793950750562288853\ 107428450672612547212661157602624214252171781968618111159025730231520082\ 974104018975922620905987451016764285904803085105067367258637575680969879\ 238949322187791309681322285184827219827149917074263467647621281104365118\ 144943962691081029405474041300353722210260217538115069579310575495525186\ 619498591779178879889747440165510668637677318975215421029103803631165076\ 920579701599103089141819437097252891995521700932384466087839655945468674\ -850 4202343692597831124240364410 10 The smallest empirical delta from, 100, to , 200, is 0.3093327980 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1122 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 33 1/2 n B1(n), is of the order , (33/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(33/2) That in floating-point is, 0.290430363 It follows that an irrationality measure for c is 1/2 2 ln(67 + 2 1122 ) - ------------------------------- 1/2 -ln(67 + 2 1122 ) + 2 + ln(2) that equals, approximately 4.443166168 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 30, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 34 1/2 1/2 34 that happens to be equal to, 34 arctan(-----), alias, 34 0.99036553533393772789 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(69 + 2 1190 ) - --------------------------------, that equals, 6.3951968273628846942 1/2 2 ln(2) - ln(69 + 2 1190 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 34 / 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) 2 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 2 17 (4 n - 3) (552 n - 828 n + 173) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1156 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1751 c - 1734 and in Maple format E(n) = 17*(4*n-3)*(552*n^2-828*n+173)/(4*n-5)/(2*n-1)/n*E(n-1)-1156*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1751*c-1734 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 17 (4 n - 3) (552 n - 828 n + 173) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1156 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1751 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 17*(4*n-3)*(552*n^2-828*n+173)/(4*n-5)/(2*n-1)/n*B(n-1)-1156*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1751 and 2 17 (4 n - 3) (552 n - 828 n + 173) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1156 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1734 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 17*(4*n-3)*(552*n^2-828*n+173)/(4*n-5)/(2*n-1)/n*A(n-1)-1156*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1734 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 8866618250175309761911742944358545388247952227237642518853902582352207348900\ 078085936236773044897981761253850932049604438341634797108011127504478669\ 981134441568064187297959623122735877688082205944902435992719383846697168\ 848529199259489862808945538529902550828550429831398306769311894719615083\ 862228349922657860512955388496282020188476729221873832238097569252781614\ 207782780296304548209886546616921530463502326853305244267411893243654898\ 423986213793921825888337719589558162605249626671038032084151783544232718\ 090575128662580508627725025953090604699613657732696349711289324131752289\ 228630034042329777291496067209250586328678763134113963197800692902519939\ / 47508051293697404129327137735976834347771551050398340753754938 / 89528\ / 744022636119158641730421636893637093792854598725903402066350650692295942\ 816273229636516017912948483392996875637607541474600205796946002952768039\ 741353494580545966103327283060939756390643412818111002418530624750294426\ 802362766848919985885938974837232032177674346505781010358357538745963239\ 114768722783101707262233411899116504055951564974639331709721241248457456\ 350484107791077397696092962376902520178874401769911212684208207282661482\ 890436066165273313087031338139024116192890127733327803570153082624201403\ 559928257454938705943146830269253094402588334738122085134405844269441483\ 381543278366092464494950790435011753993520976504678985609393509173674348\ 4517533111314134477768866521743378962237299975623281963809125 and its differene from c is 0.17790250873531082900821425108087629134759104845548669321371332333722904593\ 045509928864539396348901474229395350774148241106397634393015717562369507\ 839392036364233823720641495598734535839772811702429286751574356483388987\ 527781659363476682849510378280786545324378752135253066473224015055724529\ 739596389598437533395502298261937578761256601708125991625753592368076822\ 157341653751662983144514860460193095803149169285876157654780674233822400\ 573164382021969973223433322374400067325389173975959433646209850974420681\ 913465577124675794452488434562073502865553355108035965168287649414653952\ 264012601873374134640380909532275985342627997778566127038333363115033575\ 017921783662508526646880573686407583019721835799587003349387434344679067\ 916624205197808875803006764428643246226437415885162298314193144473830397\ 772007453923535754130280232845070787412231370571507567277458349550966629\ 512568686493184173058673109874331141342939865011593405657728633993028247\ 341028743798237680150237280732963306086484429957281118506930657410891477\ 196238842466849233722496499937914026486852032093032097880022314232512154\ 172040163335568488475088776766396917421587677384806456862230676653584105\ 740930228577684411308901396594063367716134895436231484848972486179383053\ 737288015153785132353943407560642098791632706978838773567657503854456214\ 568653987489780476990155035708168657322924416661657055217642324851643562\ 912964407869301218766725450259922394823624524925155956422607843182830810\ 595390845235907767180501325045608322351796382199084537415488912640377741\ 605849526575898810792777895571322181802712253606465170362802324210334709\ 917097968198792840348858532350074059245478393505694761245149168920083390\ 494224634053191281727383207162280479842968398325858567443755457813326492\ 803270390785697450969288151981331708534182778687464524122068677576908706\ 087527326283187423363632951402940618647531902797162462871816182954542704\ 268567344324082877818719765001583027162184197336944538511640185530196379\ 978090468626584910094594163856734129061208603004227112907417625242882459\ 275342871832650510976155700819252374505444381914158215789934072959431577\ 229648003779484712054992091817347492093194612648097031990514294554848616\ 188267741313534052158326944891717102016683272004509807269212877064473110\ 037942909585386389910680254192318397483908149671949485975299423017550983\ 696734372316098039550820860191014345221183992877599391177719464163139991\ 485403421751255154915979161005893975131305405881887690890042782176599722\ 151126253406632063502968822539938226163459916540365806211119085817831131\ 700126292353698668926234517650417939251515541683990285967937438480126999\ 061327787992917620687332024691786287553145743696515706481052471363502813\ 023596436713635209040967382888831625569137905526362587742328582907566959\ 760292633216401188519245121989668041024532180682581875363569852396447740\ 058386086777565109959858261592793127091438992196786110480359455965825654\ 067252114406340947209668848112977959115781128852962098885126719874276633\ 240888266115793096244681442192185939653234547039897657983901025038398064\ 990649440185406568916361934058712716025769939345907450204800624569273259\ 457348584727730924074152453276148261882434687182034713621067131898509569\ 401714698935211488676153995444264402369632519622720892298856084641714390\ 915201817079566228161076184366333885364011306976285991955305947626638317\ 792025715743551235043969915068398058847156096434389488117924411955159105\ 694837512852133437793130869961589984834259086598571435367774938293929520\ 789579141838064650933425827525135317325697207980681953940897492040194350\ 613216230413952409410989547545774018688183493574301426937108603401771358\ 190821243583562658136378225018951607595900493481336169358873392317934547\ 244684995756422339458401567648196410520312169014530057301026634023501678\ 420733589289335187569334388656133920004620818900511896804858419973087400\ 250199317359891458392060172489505543948676540609174334606957895286251799\ 367430458068435656169031132988300027655295658198686320717027200768812872\ 073487975383792794299175350066866727124982457092471363547118594734857296\ 968217227873782328976992745567670392857800549975929384312857271629405610\ 606780682781487009049696398588813358787040521268567341733901076574837385\ 058420039125168318202139076006865112431420261846442003190223222581579997\ 299475281933798038032797110043323379230075267590144388507187902347891779\ 475734580882690729858291965669879307830300647154065073602454066980411624\ 954143826232440206390591513302305302092630282297238172019804746572177500\ 825607300326954267394226764528525891955823750669802584328759419219243103\ 283642915459803608169884033158089698572934267015393182759849785590560699\ 879985519423203352267701719340172318239001306384056423070902347087436897\ 544658441211216392789144459482523235900483422675327337553890942089174904\ 653581610353502305749696292641996824069570573685385276795401195873989380\ 953762612396729745364334377485825236558753181975537332460522883748837914\ 238720922273735545689922539412358757694692428319939426704533393682337857\ 993569008462831282572269921842590878856869763725085991287101519563214363\ 650846951028107913781191126016313089196583850593070519822167794045877470\ 333759738327593516393277381215054685602502503113565704872026755900907254\ 812610786314497806594804835014726341120470921811061417947971502719840444\ 923148846129541232900242593587043067700681406394030552150656705379379050\ 762668160909088204702226895175130070027259396389764351503719001884499436\ 216858490219007106376329567751826000019406960234799628064121439518921284\ 455299954172276968969365101482745481836628099544616789439420376964253763\ 382934802568480184890364272318013238463007300867742569399689703045784862\ 861266357532737634142950808282167784419298559516389755317170324081988642\ 686143161015127519697210029125317022884994282133575270277524830832469594\ 312029104453889098961606079338138489150639642988763807295498655539541564\ 555674439762574587806324191333429781687501090804332302596234316635576825\ 030355106568610703870805045821149394414824535760338289974596470472883399\ 486681303582454628537593136761413709983762258061644082305831564749086751\ 235040525965069835214314001460672471309212353882519078826813080746533969\ -855 78849859605795170953574 10 The smallest empirical delta from, 100, to , 200, is 0.1957593718 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1190 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 17 1/2 n B1(n), is of the order , (17/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(17/2) That in floating-point is, 0.185350049 It follows that an irrationality measure for c is 1/2 2 ln(69 + 2 1190 ) - -------------------------------- 1/2 2 ln(2) - ln(69 + 2 1190 ) + 2 that equals, approximately 6.395196847 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 31, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 35 1/2 1/2 35 that happens to be equal to, 35 arctan(-----), alias, 35 0.99063619619046697900 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(71 + 12 35 ) ---------------------, that equals, 3.3532818533703901973 1/2 ln(71 + 12 35 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 35 / 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) 2 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 2 35 (4 n - 3) (284 n - 426 n + 89) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1225 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1855 c - 3675/2 and in Maple format E(n) = 35*(4*n-3)*(284*n^2-426*n+89)/(4*n-5)/(2*n-1)/n*E(n-1)-1225*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1855*c-3675/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 35 (4 n - 3) (284 n - 426 n + 89) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1225 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1855 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 35*(4*n-3)*(284*n^2-426*n+89)/(4*n-5)/(2*n-1)/n*B(n-1)-1225*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1855 and 2 35 (4 n - 3) (284 n - 426 n + 89) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1225 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -3675/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 35*(4*n-3)*(284*n^2-426*n+89)/(4*n-5)/(2*n-1)/n*A(n-1)-1225*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3675/2 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 1878709961743565379422781893630644808108055401491859739155250463448967049829\ 893007549311456378343119895156202368347121880378518428518537911177475960\ 160234008076473975801423385940696596637988786147463057647676339756416180\ 603397215592658748554794512463816412940436776460901512633898613630099200\ 282233440064737581570321965927365408655522397643359578902864361350599727\ 054352849022363210170181947255353517495420657705410011286435244339380531\ 742496580476219979198795057290654239267563703812215418079542182235056251\ 856508358907572252662127580036683863622461939177920248159516925322568509\ / 00624879333180743 / 18964681171233428526542302907401271867826448295080\ / 185930891678539879140896348237400576156297523050130295297362520662637941\ 471102183505264190336854239206902420653529677499109775370694297426064952\ 940191115017766592710669598934975291437713925710428694643174219221030471\ 938850866002415675507948362172493223581700801977977983505976741413312052\ 356991638494220133381059568236904279321621468970660371722846337726110666\ 860886421783005696031026775779396890040321323790681843273974688790278495\ 221605392293097592032320694978290324289743026444551058252618940051229536\ 4864338582825208516531231670705122972645120 and its differene from c is 0.19322411264713423962554265164278637003556145435631752629577957163940311643\ 012810164822305499120395327452129981638799341005989643520885430203003681\ 738297762421364718082849749472971732699125690985779330606644215173448309\ 708619172964959148198877774838217525787908912244545891341458246013386587\ 436780344871517607451178121475034452561231647482242821318959024325181928\ 848851502139010694651115090174250698577741748159442041843841302213587430\ 582375034202885681053180473705767527191264977404975374475531293815506371\ 723419730556799739014943268999030580553077426130916797694275334404674939\ 679678716776380566460780939869487199406119872629168029198725362805165319\ 317580537386891928113240730438171062931029888726539955009794178741219757\ 464909343346349732366244258617572197568181797641050010029912840043966307\ 060153634645899943111516593211287340035827271381671609274015135434625867\ 655016990874589806310983806274265415707597982216139430178144773585077664\ 771041251647315260564111757791191377795557159751181284874875215335064162\ 369874632314737957002652404293511040378432610195011804295252348437590605\ 931758442827206987540517128223468165307749378351497813715527157728080070\ 359367476140931711848441488555203255421264417327061395506590985684848007\ 726503837409797813341575257375837180678501115367882979569147328891944268\ 438423645095319880656654788912779465795563658005774712024350044370590579\ 338780749173629054315622358715918007260863896244314441465728882414126330\ 886029687906492962441616648050532688504170633449572588920392437319663631\ 914647471239245905169474252646694116225025895649298631882471942798877784\ 962268786202327237764984816280062736565036491103398300870639807111357718\ 902662822914995791368783172306633108745607975850237068041686426372630584\ 325591675671063210360475923546753224086656927190838014815407282692283327\ 676197698418625229099400189042336308934163319001927035346589220740279576\ 760962844444459947186511773927132044836455453538391983799587662226277457\ 022180072833212741809868458541832516534231484493161304894104197001134340\ 477684069678390673967751447257953641590244275504419148897827113761256704\ 812226931097077203663914391481543607184167821617588630458947749455448407\ 476601520326594660501061819134199623180752709291854180989565730256425606\ 234720469090301624508760093155986568189764435285862189197926950944242840\ 383685961862369641399249913017411954331123573125935679142911327934688224\ 754263547957722324798398313545043858408442937708236229615026686527405992\ 602741813427500090955879236051854941049226187500293168417777422123168912\ 679209983057075188668736556578716265037366820721416439735227862812308997\ 158765225711509105161221511391104322052093992506474592055646564970291579\ 828327145516197005930412688207266412855228169258379651227486315538074922\ 475439951522617837360169665458866575328406725961125602832617296467337862\ 050827523777933828150528965968393335722701964599579444972890274138579562\ 067654782336017126961508641697209720932043004645018917085366538714015627\ 733513950873396530989708343820894802460290760661145562592963487289693020\ 944602071303704115310034077155897557611278249458808150301111745503606497\ 153485608632266506296233143285241905375430390449873781372157038668221703\ 032015223485505845902803704359529938190523726183892513152981372330140752\ 337353156451831439497980117909502728407332575581772569418281787030895877\ 760541080892310463948302014331858870174151609177329542370958119649644201\ 012244600909790228154705924413217684076051248870787684139069693789876180\ 279903042459170369375214533742515328614481458344402286428705171526715681\ 168900172975766175110723427240946027353809075403183706214803125077779074\ 768367818385785412127901710667500454023969862984485416020770630707671961\ 462260445311079393791078999949442473154081346238996831744796925553502994\ 249533931134299067735691501354042721311111950015472393376276223564613568\ 518447097819159954928462816235114412380851574656548748659262274534387077\ 048042808799230643589015647764429442920403216210078250763056935022480567\ 142136716181132867935289139416186465739606968900358359900996879745392553\ 753273849496536212812480926172943680442400008335236775116916592134778068\ 226123822253001273664156089802633197573638308881249023906373892078444748\ 332353414753116855448296792771468129739682366178434021679778331624626443\ 354806645629909056846388734086905300705000756867355213800912196357398638\ 082654141694432674743413092677734331971607006055224434672526149710824318\ 962294878652709274772189542424555000527523427147532430363310014193703968\ 930942784900264513232361691446619832614755080638434888898498593089065952\ 284365541139573298558391810290536861980475091239200256706364959812296327\ 192536758050000996738236958671796636892897465495669914491966847937756070\ 243085356407016280533312604440481114499520549541936970609939786920679017\ 872331054423389895047889481975500784180294167160850227027187337122288667\ 605344319870806641525287716603709162462085718955604047219772366872404361\ 810164509197291450642762294955001059327348655294764394590765652892405311\ 395164395576841452883989031199060016098210940797478096925547571670966924\ 982908340677482736993812773925685522009596235540251900056100637137905920\ 417331593104418691117862247790134540878989180117387005178248668801453409\ 296907488853303268774777025405203977151003130851787007990401523621280516\ 152823169301966218555574559176618091768147208818343440338933009468219686\ 353810622141589683073985721169587128760630892801920629484262964552708744\ 795234600198017603769699549031229382963476842467062558906297714900572922\ 521339256743827523797460640119958476912632471325976497387191880610278192\ 611391220511501617080143197080903456988206498465679330657861670919506725\ 376282816777566358519222118882684968274920030877337175911469448763911580\ 318883365565377322626909812134992634245412080711750599964302126922089687\ 291962170739735418857512450665527830761921830552578240659577323735660170\ 885170593162490223433547063741531714023709909444957034975922885591649039\ 166537347070165110442717297777218160500633847645633117450328411263350786\ 133310460058224528380138631402025479399665627281690095631195709179119351\ 051052469655775723370611593716804934727896127936273051477772729700971433\ -860 148093933608530354 10 The smallest empirical delta from, 100, to , 200, is 0.4339292978 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(2 n) B(n) d(2 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 (10 + 4 6 )| |-------------------| | 1/2| \-14700 + 2485 35 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |35 (10 + 4 6 )| |-------------------| | 1/2| \-14700 + 2485 35 / B(n) d(2 n) But , B1(n) = -----------, hence n 35 1/2 n B1(n), is of the order , (35 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(35) That in floating-point is, 0.424938573 It follows that an irrationality measure for c is 1/2 2 ln(71 + 12 35 ) --------------------- 1/2 ln(71 + 12 35 ) - 2 that equals, approximately 3.353281306 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 32, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 36 that happens to be equal to, 6 arctan(1/6), alias, 0.99089206448776102968 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(73 + 12 37 ) ------------------------------------------, that equals, 1/2 -2 ln(6) + ln(73 + 12 37 ) - 2 + 2 ln(3) 6.2401149858787930280 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 36 / 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) 2 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 2 18 (4 n - 3) (584 n - 876 n + 183) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1296 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 1962 c - 1944 and in Maple format E(n) = 18*(4*n-3)*(584*n^2-876*n+183)/(4*n-5)/(2*n-1)/n*E(n-1)-1296*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1962*c-1944 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 18 (4 n - 3) (584 n - 876 n + 183) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1296 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 1962 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 18*(4*n-3)*(584*n^2-876*n+183)/(4*n-5)/(2*n-1)/n*B(n-1)-1296*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1962 and 2 18 (4 n - 3) (584 n - 876 n + 183) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1296 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -1944 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 18*(4*n-3)*(584*n^2-876*n+183)/(4*n-5)/(2*n-1)/n*A(n-1)-1296*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1944 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 6340166531207508261298044323595306664513231993880545163421755869414371261146\ 213811740243465910756558123724032871295603215371874177028161732809307122\ 332883237854772402291263098368571650157895613522209780837266713201737509\ 638175255251609298008560615773803812893846445128899572985701476262197914\ 450244383347495528893612346456365376032517703753879981692180481807667927\ 530071131519211018583440219462201476850143557865098911921204828112406758\ 940776024586326256267024826581595176569115903236146985373194316869635905\ 510738822192482068990564750899040282522324230449434366491036471920570516\ 093237877258888788420587943636478147783355098927853087940586002117202900\ / 4533415550952161322737065519396290523160946121760922085194279632964 / / 639844313869345617006925451275408146273726121402475290809612117608716120\ 070361720517964799215087586516966333960162102328975338517112261774969504\ 957816292046464595603406775580990468671595018772240705224668735914127652\ 517095552171133229026500042600304922865680538517445828505581093936678485\ 710044288848028299078693121924831651141078614989783624990764685770567135\ 484542718027315115189754932701000106795588053275147061640265036181266848\ 540400489883047852930014324060338996974529712760138730250400131065390345\ 087316321976282423532576961347467385177703281953836689847419769882389916\ 073113353333208833009280391586464777821905034843400873864427937080437274\ 29434166859066444404381139918901461689978798083859202286097115700164875 and its differene from c is 0.28832040497006253455357169329243556047511629393354268546590580174359206077\ 656853181058594865454595230999437350782559347681146386771238072265954381\ 085684981916855799829393986521989233183795960545541081981051574506135237\ 447849263622175692016899326350169584172119722145171440962813339674921632\ 257552764493135898133191682892914299555512963101040912784945887969500474\ 578071971055984424562820791356104238197500152205263352482340791270859173\ 556263749364209895982318327421468840260632983844694891780041218688969941\ 249604650457886617559855951533118989242833786830747929538267302075073604\ 108256648030605359480327988668061671965177862635730248967824153098332104\ 725433622281921573580804664907391392518107481825128454920286482745811124\ 857867825422449008589987982406699811270225086661557237956589983072465958\ 276202418282386689248039764015481739723497692538690106407620263767304894\ 306910059482564158579775817132085249209354597059311511917315965323885304\ 377205244604072261973994318967713116751104343786987165712001906928318053\ 444489408353643577276951968181729292546653002427866845617154938628629878\ 017787968774717408935919822564329422920875554645521442127163643330813045\ 677473419259685727410848992545710337144794090451988522630246549175804999\ 660656400656348798952935804641158908332078001465182810461882858465457261\ 161116452966003124619014149784132752551925126062801860289738659249706771\ 966798326386484778833143433236697584977664048599197086072212173833052516\ 177836063720127030456278059625595565579172393526573546448552713034948116\ 536686486371398448901060385678959948675197345981326531305593858330110537\ 108283902458971319139285463753052906666627846085920452345798878298634998\ 752034220819010884271481529648372019934015011767380558377586024294015860\ 475582948655799881974708941648947950223148921550719123945375227557056868\ 958664445422661402365119995161183055259321774137807479056502949416082502\ 568118799164525726094217275618743341669067125969408571025998323168919242\ 115435870633584558110738508568815375230414567347613897456537787292087152\ 087608562541836785057509853013030927340201195663815987484601720010462066\ 502668582032765526702575001510996439209595689833371870964220621227610804\ 817717642980229718284231483596298101544223169060792139398467570981006307\ 945947208538590648950652652418267214844148493764273881002766386069237791\ 443594581620386386057389591306388916766730804244522840786945703771640858\ 529615839352391414959401412257185317363840176337820699263015032570648850\ 784146759842728362676382848747703712392344560565971671180430292221795499\ 986114440942595034810341854037790868515750064901509689282121258765514700\ 092177340823896329003531146463512473838545066427472742245551649751236461\ 551977653747594342772062364046895860050565977678388146892033657082153877\ 058765289474538544976472506527804798620298039196591818423331340197644887\ 431072696285594861735460537478986758750064078898028053431958781991440973\ 433722140740923771545758231761940287386821212098429809585980266982544807\ 686816004999702937265035765131702331120027565839850209845839151720890303\ 991710570789706113150579077850496025993089336305838326799265597224806226\ 980623982941405117343380757571906213652502378384699186903910850166112517\ 523182057275160048648997014233025483932986897212151339638538654425572459\ 736607267202129509190141494325124265303726416829963479507577459559781229\ 723280419053126638911875881195528269436278381456022851728502856172731638\ 096937967880610775856105255045989879492695253322135568031638150540320733\ 135854873300842633548871312112915654575544694510523293160343963469089283\ 894346721060665639343524053332446663380451131991636059317020177540385462\ 352521412404211659104470291355991217570347977711058422979502775672828033\ 966758341952193530085955895554156745495610716163029921779991899220981617\ 137621828880780880315074375447233834403358458372135439888496276883672970\ 469348242726598989873719260494011479078678233946828691395140672292085949\ 728438320229314990627684794277926247532469197127061865168292497139061490\ 269050802476087406038874655707696042605699772301383089084286338347559081\ 002400841698877713570568721740120228782350140773211164584588659325202751\ 943605903605662168883486825752328224952447842888771722530626850838733968\ 825621109079255175763704900247881549017522675153997700890398695008959161\ 289215332796910228408253720329750028317556932641661182135243068713201342\ 933376695206013297341770049777134620678700902285265616296828200612484246\ 041090255170330254760388145248488893696709428073827602745517625623424168\ 883390548415659521821664304586485235380275877772729676300952988057793623\ 207000044220183824086803241671627761319297209541807699781330516284648562\ 140651527455407461596234217213410129563786238245775680761660690172175347\ 239960108646268428661319779648050706803072871649908652581672414839987465\ 876427235849694752407575349627240527488238773341491764296404548935816699\ 846026771187502752239436821680620311531220612361314047242359516648044272\ 048468114800954104331888779499042238752632051874180253985864012481940734\ 016352623635491311295492379104097096986184628822777371438464662000682662\ 794991010599190601750291991595116199633325599527821356461898874765131335\ 128331799098614524818163721552254842673042477591277609620848160116199185\ 717140632191075795990360874651911040261249842025233943072763651471666720\ 236284671787557325772776018053988981112603226009597607089348127145633991\ 011992638993271238240707715174913644744686944204101959037135576355001034\ 620085301700764942009246744137520114455696949515693271306153744063427836\ 539853391041870930140132992039454881632160224731297973838324687058705052\ 991847985611902319167508443053289515876367157092037704690180894214896479\ 370724528770781388472151736689204052495006838282116532901079829425280266\ 825397315239371217513681081170149638684388798439893377200431642835725663\ 106574564604349114131172853111350293500716042399177471780070676255211226\ 179436227058111266553842274995465196441573044251950194381959278022650804\ 256109722594669927202412028869063788488751198311140641996720760883498978\ 819223722290302942284371616536959168210249412971505648272244355563317008\ 845091357808311371935912819340714669646647747789926862865582197191286685\ -865 9104169201063 10 The smallest empirical delta from, 100, to , 200, is 0.2025634990 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |-------------------| | 1/2| \-15984 + 2628 37 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |37 (10 + 4 6 )| |-------------------| | 1/2| \-15984 + 2628 37 / B(n) d(2 n) But , B1(n) = -----------, hence n 9 1/2 n B1(n), is of the order , (9 exp(2) (10 + 4 6 )) 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 ) + 2 - 2 ln(3) That in floating-point is, 0.190835478 It follows that an irrationality measure for c is 1/2 2 ln(73 + 12 37 ) ------------------------------------------ 1/2 -2 ln(6) + ln(73 + 12 37 ) - 2 + 2 ln(3) that equals, approximately 6.240115782 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 33, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 37 1/2 1/2 37 that happens to be equal to, 37 arctan(-----), alias, 37 0.99113432072345617754 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(75 + 2 1406 ) - -------------------------------, that equals, 4.3242396092923923194 1/2 -ln(75 + 2 1406 ) + 2 + ln(2) Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 37 / 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) 2 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 2 74 (4 n - 3) (150 n - 225 n + 47) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1369 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 2072 c - 4107/2 and in Maple format E(n) = 74*(4*n-3)*(150*n^2-225*n+47)/(4*n-5)/(2*n-1)/n*E(n-1)-1369*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 2072*c-4107/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 74 (4 n - 3) (150 n - 225 n + 47) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1369 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 2072 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 74*(4*n-3)*(150*n^2-225*n+47)/(4*n-5)/(2*n-1)/n*B(n-1)-1369*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2072 and 2 74 (4 n - 3) (150 n - 225 n + 47) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1369 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -4107/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 74*(4*n-3)*(150*n^2-225*n+47)/(4*n-5)/(2*n-1)/n*A(n-1)-1369*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -4107/2 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 5774931087404041876369324416669333128985963906271011426813290425712300618224\ 908905417010183689794749467968616660273525355561386936406568307177592224\ 503016326734402644556878104129075826944614942043743905377722453632076413\ 883678548997131332629743920341467399708481715759326082980540275682990018\ 152189414056331086874349421024280875178500149018772381125295837982511281\ 249581627773118617115220235031090358491394381106721828879102319255099095\ 828647576655085193929590819122776832525593979089327267741174924838905112\ 290600657509425078266535591867852783528444932578812208064554455847473985\ 341274371982865287983711339449074797990882814628587318953074174038149800\ / 8045565207 / 582658774563346870885996870634715574872972571888888858606\ / 361621303470524063044710670108206177005770121333197737820462938052791813\ 789189601016517910782914887672640896085254516433398659001134525921267658\ 510038778582718132227943176196451855141911623334154408911097798784861749\ 979695702285820919872519808776311663416786665952620800854175391258322457\ 145066310099397659410366302248138455234629118197785702436120401065972103\ 633092542573466543554044375368919700927667459243665130242203241554201037\ 182775800684081580840847980495627485732394056658808258956286538997284456\ 572134416798356664940095524903564663033447200742947509027404893637465728\ 56757009092312281104041124125 and its differene from c is 0.58098502144266732126159776749464335092041253161771541388806990715911227424\ 654110228113429010579761366781160501977882033500182557816845501522357870\ 996888314915433351136968660440488162687987095032209281502614592654496123\ 602509096895982338698712662614316234128098304625889924545854723245055415\ 339591669905845251450054814187634507879062020398959558935486846389961262\ 852614448498153521803240729316550122278303445375941093898307889417586948\ 675402064604583574479900293636968033801982855721010160513852315613539432\ 002539409535037966042955903720572864363000901681364860419849009335008955\ 519111771786831323217489071330688341289687453695294765940677501640899537\ 210087307315132076512387742559914982646316669389163880185463874503082809\ 223968970842900957149199738070659958173132285716893401490838619926948746\ 046570796253710329110944117094941523227600360961368749230606953113880664\ 950123176830155419035019799929192390419165051080877208959197324637628254\ 841934693894355171804244017798774453328582153349237135337203976876117207\ 164235934694081693909021076991857116792465390072526682223640327751260429\ 580283325361922535127283922448503559555862396082320289126350952287021339\ 475096467264422451814858294903020456513669473422367381155506964488601000\ 643894253105006148312488858526728406248776995103309385640023776283499445\ 325896392821998613119728056932966930044640943986906409561026349646305594\ 489276395715168475568294024280415904836758680048326770661787898201965185\ 829473838264082607378749766852156705761369729302596035969493175206395339\ 851927677610183972320527627232478919654238991375911591362394188373703727\ 032695233619926558074631809834978690719393659943399214917996831314489628\ 733334489982403212468327825196786022934081608770144884100070700892510579\ 245337671928884119759769441504306991835514949028123644912286882760641196\ 183969912317364393008268208731731871370499051439813036008074921791062316\ 263163983269740367538316649512676519577909747720842531925605087051472970\ 901006212413133657497490952425240064287498350120945568745045055871462128\ 890166095785728191758805308336566812119835248304623187184776485704647899\ 844169589085015610601349144122975474395343977260109616341324002500240139\ 047913882010877998702061643946986930574930737801934769595005331713986356\ 522778307696407880445713126143587300627529666428119080194451798208371571\ 456615362174869885844382461510937340691819585679767987481976053805527511\ 404951625022256067999065149148813121927748900243124175690030764342211552\ 019544150461517875638126230687042239480054392175640356163162313486155644\ 547093921764196237387671822328887388315501328214958534318908912506187999\ 058961484479040780005005709619764310570781762313828658053323892647030769\ 640163717328061021453015524283990993537196149304260992542115250797551004\ 959510245090881639168791419790805127599884987157822145076326125278513208\ 668460319496449682373578472196239049346534146856204413069902231659476221\ 418565717319654182383578922222155679377116371337955154112544843848390493\ 726541667835720022451855741747031698864774615894470497150485489170623116\ 267969383403780697168722940465563440663021635864330336294554735712195190\ 438959186196140555289072086395400528031056168503368630587194528402652993\ 242988483001660484624449450825903147546132906581708290434928773501156862\ 261253314130911767425439650857592046090468713418649009759457192337028071\ 098077295327873511487467744035321213793477202269262651563459501870384590\ 960909466996035597053048406880216363971188776430564507178177671174758777\ 091855159294539393516591778209175472279526115764934351275594273386069611\ 217050619141883885492558621925429278964203505524532914394125573350625775\ 430686362856426954500152521806871267023175436268145511510325936711699806\ 123333756816649464534430874824534794738749341368567826809480680112229090\ 997846448640462940819905339581259963421186761363184986294943315610260329\ 638155831408193452156175404105582512242801913340135192626939021558692691\ 943559732184757349893670526693226629668537660454315252755539583285110684\ 908201791447678514732450239833052391952374378934786969573177935207568421\ 211965485968124734818618169115411057729951108783446023615618285384416749\ 791476987771751732649417386591224590300988539887963457775087938421836084\ 554437006988185780860743387412548575489911952822209913775075767935856880\ 976849645609917613515225869917458832388074528236948073598742521655340453\ 714334712439071228918939249318068545369305014169640381129295731210847113\ 125678708298944779967342571444528998763272554039376598890575635350158144\ 041709699417046748720475441861529075788702885619696552013709817297526808\ 392426565147514754208653270690739390435351306372331023497218948272419039\ 118955379188439484840951240324806075593526782545107438175487023527682499\ 892410353619207052554105169554710394383514098762416406540510652464690118\ 755549714336058541719640149653138613621093534956780866618150642289353696\ 121330749108088798557102034943294288684276818697014484130112617907011943\ 427675506852592730711353593033824329930276424500059028355633641425020014\ 061791483391575034807436165558489550114056321005696712159788172546347178\ 051974585532628752752504307994719079006352614070073503853823100320091100\ 655194633411334037645134050055216420295405884097083394171179219088628311\ 354101669249535457237458456596049902194611526090823800520916531694338268\ 969217495183334583792943052601332061804984839599654179689737192812288671\ 066109013008851836611947881115443727282786728330847450521591911691380127\ 595014593385469298873067611466133292865328062658717803088026344373430919\ 870757010666481437276243254075437117332266404019896307009416075285125214\ 797241630392201600137790462599199589868620156799383803545796842254269755\ 272258576859405049884595730224601082185792692072979764472576445015950326\ 727920198264156368150913119004277559671861186009790360265225228077426445\ 979910996241942541261347190984776346899126834177441923555127328239029744\ 450260130307212752969640490600011901258290510012677014685228627966257440\ 420627983564982271623077767921174718964480972210548548599567204096230532\ 165810424311945691634270875464789644240188791855118727211982709386796572\ 950920071681447563955505831674206117572648063828606908325980619224461939\ -870 85879989 10 The smallest empirical delta from, 100, to , 200, is 0.3135320973 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1406 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 37 1/2 n B1(n), is of the order , (37/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(37/2) That in floating-point is, 0.300820673 It follows that an irrationality measure for c is 1/2 2 ln(75 + 2 1406 ) - ------------------------------- 1/2 -ln(75 + 2 1406 ) + 2 + ln(2) that equals, approximately 4.324239621 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 34, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 38 1/2 1/2 38 that happens to be equal to, 38 arctan(-----), alias, 38 0.99136402303387156407 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(77 + 2 1482 ) - --------------------------------, that equals, 6.1030672323035062792 1/2 2 ln(2) - ln(77 + 2 1482 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 38 / 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) 2 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 2 19 (4 n - 3) (616 n - 924 n + 193) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1444 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 2185 c - 2166 and in Maple format E(n) = 19*(4*n-3)*(616*n^2-924*n+193)/(4*n-5)/(2*n-1)/n*E(n-1)-1444*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 2185*c-2166 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 19 (4 n - 3) (616 n - 924 n + 193) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1444 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 2185 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 19*(4*n-3)*(616*n^2-924*n+193)/(4*n-5)/(2*n-1)/n*B(n-1)-1444*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2185 and 2 19 (4 n - 3) (616 n - 924 n + 193) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1444 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -2166 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 19*(4*n-3)*(616*n^2-924*n+193)/(4*n-5)/(2*n-1)/n*A(n-1)-1444*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2166 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 8350924567835572298704994025670359076752971687080137822327333963266138461278\ 467853072605679512521679124357951165199253723886056601423536541259396825\ 454484931229808132662519127844653707945948016367460673004980221398133588\ 015025432526088217727557735908868844307246701361366620149452659039633747\ 328678464071874231990185723268044988758520683465925117940715912705926683\ 636036960518895283333663312613335649242288999923989123502850947104346200\ 088269981565719317580651864380986592896853255923696542992248962177054970\ 949381576730410503071153638598181632533275678380507624329704478713856089\ 934623503472010381277571611167006202570481022416173696523760621505761596\ / 371833246712150607897759629519710293112390022318469186933900335602278 / / 842367119827410700767054610508668354210791418277391526249124434840582133\ 646576187212082751797741896867871980349583531134988096365066374507301811\ 637059892532376728949961897545128583918444867960479138011300921012914657\ 972853508055399273182096491214965096334519715518770636608331659960352474\ 645230308648088093559684895770186623691420560934359929231921690275816158\ 159978267415675180290943887859922419123001753632537658678958262912198000\ 857988127183304119134700992365338311390381378076637526846181782408877107\ 214343283110183047395561739565245309303383540498022840457324310399119825\ 852756628910366475189712859674524989709222884966210654972817406816298609\ 683219659782594742548201475814500886445817499988829462014981547275466212\ 5 and its differene from c is 0.15561803197318727859657475799121963874256701544383046474939565348119965541\ 437001573531161437975446785401883542749161508110980354836810088825326857\ 029568577829565655667033328948691271851256794446101391583245681033516035\ 956627370422689024328745415538402371299298515981059848060005030672919482\ 253760382479035746118363058287940349709985407228450520432340280817292982\ 455647860970680442405161406202704928598531102941279365683373701952980625\ 223265902857806794895543680166248943933915899583449666622491561965779226\ 172901082445666041825576153962740178783099292391072331984675199592408056\ 827918652293027748722873697234229543328040938221225771312505842671291632\ 067701842469054970511975901115840643498966051341694915669620431001660135\ 451615731626426934585621735809241210292915547885555013666242657028063354\ 352109738675568655685955062005914622550439559471014918234958449899266228\ 254099319898062263967624293323239057794046794705737742217846099728998503\ 362772508152420246110740561580274115761946261657705220869655677880453978\ 749921611198209254019240371376269869935470349895677880960133482878334738\ 044731631824127058646358685301208394476571030824970153451501422539212737\ 547318130941735987242462350820255918558805297314475557086184058488491405\ 668173263167360359615341713407500401599308537984119535174752824097075946\ 145994048221977390151645082502337103970805701068760832623612280023370747\ 229331853235460124484929564557704903381145563279937521687923982956961053\ 258866604601166572542846050560316576259962534266686758439250485020056514\ 982311261915312930148878297560016221854227913631748456773816968713130800\ 402458184532270531687395033663776118054975534258204121575788208164102275\ 790949776324165344087252106171247741997584743763700698424589499775606533\ 435557298482410394327657581667720880206459874143842375027394898163880030\ 797562898896604971349951242181466095105801655895008043520928163305372707\ 380073984102927021127870744193408839647533444748137740533279678421499741\ 335857362133677270172953220067388597680764754708122577547546832023469589\ 228169820925009353302870603019929562719313708705508212063924786531318186\ 650759145576232500820589332216233161199657938078574625931696064774620511\ 366038616836814959014972308427899006404931655532854473131886572572967980\ 270929154358007883994940921880397030797809261635431087514362402166094739\ 784538398361922951486936853773718240715697175812837913129319695069347339\ 351685487390391949177308265368423001923489528579899028912179610607068124\ 195651171418496507602363129508597833481466660285052709568031014635307959\ 048561041810372256847427595780021794799310419433730854681951446224531315\ 188511743249088445538733567237136543511620121733275046281125490586829182\ 816293469883003397442459166320051944138944908207099740379832946091882207\ 857819785107424687445690048128524429095559763587900578959534496938223515\ 146822101717149210735128387132094966107578805511028821481364910241229099\ 886177126661474568326381966113426454119684980008903848522454978103817084\ 027106078481302831734650118317155257534537316065394339140934607617136508\ 338413092963698585135823433664490114939612414001660581020263345540453009\ 259783080520653692812252243176641589707645828521279962402400633860309380\ 169413284648762362457875068403583169153892056184848732649013781577622104\ 250553557220242376842116788650704648913216562202066582967672932065158039\ 954046813767674032502860562862752421964117649063918799501871198695770738\ 916587926208000056178089057097386404173248288839694927439845045787764385\ 515401752164322610925906040381341763948316053110146562438567709759884333\ 741559273867242268972431511314941691028442580202342437214185149690168672\ 959918618820621432697385506712631927622654893944306984004335094557558813\ 570383572318554247487992824875016405302885753797373984508487441881139805\ 340803521176624287563891562857445393651806437074851077568737800352816254\ 471415154732260591599084300477101491090612862180027595731127941389864697\ 705955135978285347982439147627084206537945477076571034892550722592498709\ 461125626923825742832614743670913983217539352848053285263684191943557083\ 726431989077322842928322515274986801921607028211438476736393235647517653\ 235027973945387963555914450200041661228308425283419433196103261574194546\ 143396153524119402699211424262910679998217207592092965478871089746506145\ 131794741686089530798511742606872460084168933879371030772843763586176238\ 351888513777884197739905183850397199626257920904890433005340991659561043\ 445266282656212444748421989116795477618424005933151503406780104441095224\ 430072668101275030871354702117295830726088711733170413332491983393525567\ 584607467646492906861381320428513734326788796842167145574257402289070913\ 985826324393548098622137196132065467120853688705376440838557607469054366\ 523924395624510060083693842427173693114023272294496796464992525319984625\ 736090100881220371481928149378307385956642049056760468968837884417705223\ 245445546185927289939928806793579511145214466720507320075040271422239613\ 713873656980659287243869658377955742644145729416387178341254040985891680\ 085414287450327384398018593810808039370123900314558226941885743686514442\ 447544946732187441671850409693017940929603285184773941964350897192518138\ 258679472693674225614101740699044990372894190832629211877566683809754282\ 536722177730432494026822914408469499992170302103583065842085673586825439\ 260104164211937622468118219712180880436274249809705194796065496507312297\ 908062531950165226314917830611617445455488340409441315292056234562150170\ 163297832515667986986110231139165777183585153029017335208328697140159267\ 594727510292945381464747390887130382443729279803474203936694685632201381\ 306595790366471864620404197181866531489773039005757807944524434121528757\ 233120762694592518419276921965842127576383110700178777436020899481618600\ 002960339203260604511936708843720650995671031990282602269074760445207923\ 676676000015065200795541844062072095498842988534091519857673337827676405\ 271471004702857863987001384524435820604332448021600803761641555918906326\ 295947582548478667754661376219143470470278212134353286260093813993874614\ 876792399082025730934879874314790558780710245210189858793825723100772024\ 811823999240158953321185962567915463549838094831997474324751892466116325\ -874 6356 10 The smallest empirical delta from, 100, to , 200, is 0.2064595908 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 n n A(n) d(2 n) 2 B(n) d(2 n) 2 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 (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 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1482 (10 + 4 6 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ trunc(n) B(n) d(2 n) 2 But , B1(n) = ---------------------, hence n 19 1/2 n B1(n), is of the order , (19/2 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(19/2) That in floating-point is, 0.195960578 It follows that an irrationality measure for c is 1/2 2 ln(77 + 2 1482 ) - -------------------------------- 1/2 2 ln(2) - ln(77 + 2 1482 ) + 2 that equals, approximately 6.103067210 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 35, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 39 1/2 1/2 39 that happens to be equal to, 39 arctan(-----), alias, 39 0.99158212264804347998 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(79 + 4 390 ) ---------------------, that equals, 3.3060990040496472527 1/2 ln(79 + 4 390 ) - 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 39 / 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) 2 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 2 39 (4 n - 3) (316 n - 474 n + 99) E(n - 1) E(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1521 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 2301 c - 4563/2 and in Maple format E(n) = 39*(4*n-3)*(316*n^2-474*n+99)/(4*n-5)/(2*n-1)/n*E(n-1)-1521*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 2301*c-4563/2 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 39 (4 n - 3) (316 n - 474 n + 99) B(n - 1) B(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1521 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 2301 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 39*(4*n-3)*(316*n^2-474*n+99)/(4*n-5)/(2*n-1)/n*B(n-1)-1521*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2301 and 2 39 (4 n - 3) (316 n - 474 n + 99) A(n - 1) A(n) = ------------------------------------------- (4 n - 5) (2 n - 1) n 1521 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -4563/2 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 39*(4*n-3)*(316*n^2-474*n+99)/(4*n-5)/(2*n-1)/n*A(n-1)-1521*(4*n-1)*(2*n -3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -4563/2 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 7112996653757915998428028956814143173118108141156437799471280803563689019450\ 017146099601677599253064665657797244436757249176348959705151330628009923\ 684747713701921116083451749343397767288123281941364938171526196449971202\ 957974382663432600616880252343502758272090910993530321811012204366213984\ 049278002539819115544174595989150957743233310189440659496655316224848872\ 793499819846263997268146031635683709320744853022787862318036301062956875\ 395593972175117155606114755343098499678630889845136389356526648713572602\ 247159574495933527097414738778682266512063928660804943750869778505930351\ / 239679899299690738352717 / 7173381297720949651316114371674621356072986\ / 191313076626690340810019780066981698823087053288904196546396042202066157\ 931612635284834476609846110163611501163296085725406017835245802910982162\ 182037839024764970258661229353664041531498351006397123019668211193833737\ 684685628815141720323140250833536979314861219188008175842303348568178175\ 154763848089890630095718153745794720101236152502121616613605738288143111\ 326234712955988626143741049987536849965157185091323247718655034448761958\ 800048496676529867777028633809463379817012297405714342597626820821013661\ 716737950913714206986524628720290330952042846025938464000 and its differene from c is 0.54603315705452629051071213291726554503956569534320338407449354738623109186\ 526416393603166761969076740293211978770330256293864190906312266082473062\ 785954758753638883386565684708004577041094660392213383793120003702126273\ 565513996205202406610748254549532242773050167636139580255219563064078200\ 591711001900119424756186939830815166152416550774208522927008419290654851\ 567576802076428265062213371201550544894201878940943114172866228212686123\ 174720817923927746653031336923200360508942476715021644491819079480943590\ 496514723126035570715058774945533246115317173135147240057309793683891827\ 756124269282141870094245931713496166991461696472879907078526203995837598\ 736399411928072143013374687284116008330965375021431984882231296867059480\ 635898828438398531000920121240746672644070238704154372397380267382464526\ 187671739004038626247119922910226630289005421439696000906897748360459625\ 706948536217238428225039649022260874126584055350965304700054796695648113\ 330589715662920319296591193774387113357676175492772117567316101426406546\ 730729463340185478099004393230555913572839663377117661193247500802021130\ 045616533901484303487458271149769560007181502179319186540954576374335765\ 456669714922442076936767650330766278519885500632450262583740782134658791\ 569809129845107248285560489836621286745841259583697646043920633112987615\ 716177709912605209702032687091216405670626141513550846421037495237052365\ 146340842891650716006808017457627456205727244770269468570116268884433536\ 736356878557301597790549320572758054405756643742607586123129185524856592\ 041784801401044979173149595320661171686095930566132690820995140696904376\ 016220231000135372362012771981557279312832421902104237364665472620121240\ 484387258901975414333652086849170305548670789666405216897980727834713073\ 898114293824614653723602826805640636571779700600320154306295351010075233\ 176373214624817888410422414119735372864009545659348863428983329205205051\ 753768827723768054224490664158138586014679670523327905285495926971120394\ 469523827075327195278759453002228883432105476734058715105364986300625961\ 671940230003965040689639133992755504233454513793413707173158197892265143\ 843852956095039581438689832130581269605108854059323712924532237541326407\ 224442468338494190221512017158905926733238605651602903869497045907086644\ 090274154075381255527491456865160250803863451284492727389381650467649859\ 657338764918293332183178034699373895528919847673411881967553251980874611\ 032205216227024587747737557454332813051159604281001598796428732641372913\ 836780958041285696761828721898529308654123036782170344079999364425485857\ 056145147861192366095842554468311213429869890781974954381190766638755189\ 173667676908281193826737713473554005267436779566429384455721346881006708\ 070451633542162562289748175099422350028743017559234851080186628968645320\ 848933206777121400978828991911904922507500564295457387500278804697495745\ 223237931729067158821748221320970919410901170641665661792344907374360476\ 822796055731840349568472137675074418849736150937280116678374439217551468\ 840625202684686626432861202358759267576136029975558802117981945095115242\ 512123454396904942090132310349144844658821127010154407499507137734347060\ 523125206438410604912814872178876577350957414267372846534791231700106064\ 230133090595063212314121035609127349427306207187248012809215776944432962\ 796466263890738490952374349134250372415254557229933484730873311351334621\ 649237902872181651677292972751367316620641607258013824678389453175062798\ 072205598640441970207325947858864378533610660010287339880723624254961302\ 992865027691906804351901242711535621535947139507053071006272029367195263\ 293925929762984847444179597798249224790956290905933940319812946490843679\ 775562062439397699425115971767414900443982217124146585177240651429093843\ 196970482362658779544458249057647235398378120451442466219376366065629072\ 226970474366343645992576240466493868223735559685950718903869720078663054\ 017041895284555543138239995573418975964237183671557689506462059048901637\ 440881260746535991855959319154116583992592154892848299061008875889860310\ 616500822240253562470057651602591433439728423625952976973601237836585800\ 473616867017316843190256630142891007422291660149380306236325678939621111\ 918485745282829510096066130218165096253620352548566301340801399285583252\ 073951297409892915060820939514342771122446721528882381488535717347834783\ 043240728106831017410106560812122709272293021562761504109209670117564723\ 624907994092569525163627873807292572479674696421445123700565803855482446\ 153811475574759858749272576583331420743790167772418072585633282931295248\ 767659130561884149520159242693317258949665992977173608463906073979393065\ 970626950522146070295889438292714313951970393467830836592086891862809707\ 633920857665353166557929642277346920843276331146912058327952743422209214\ 906478472571042606493031483442801032988632156539630217988445959362055359\ 291585134871121165664759314369002395847398836639032730154819157828279385\ 416181732774136579529871138676911526411037075937431474341948029214560650\ 473898862475521692216545984179254963980925131765657157512690661840968300\ 711738590719784177646656853158640399178299591620369555469831549108292958\ 010874307147516865835050162997280851206300635258256937719450515198857074\ 649421771633004296678558717251900443737180007208959761307413323359679218\ 096293750657243092458367240660875999362350644471729961006819957260027387\ 489240471066684348606122562565574524298430716674356995764517381538663072\ 022710435950301026239642893396076501088064456979116033033182088808729819\ 251684716921702802059605431558248863362329927535181321759824683472868529\ 208704732573979388398451720304090507982724543285684407792562473463241789\ 138573166138310998646754469070665216712827834723535010987852559354189842\ 731575657383834475968147111481151452504204933339281410925069262499542146\ 014216198763152466702424927560104639630364299637424686165447012423686336\ 074513911567289492874993946005374058409260482765096603441206700012001670\ 168291676841674379556564837018740291371633501736605823984183760072629829\ 374448754088732196998984359950877274944977608882198948235149463335837002\ 557106719931906225717661708212579420683621367342614412968345889894397881\ 48237090726681945062514209293139326330368172523043522790334440112758578 -879 10 The smallest empirical delta from, 100, to , 200, is 0.4470203156 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(2 n) B(n) d(2 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 (10 + 4 6 )| |--------------------| | 1/2| \-60840 + 3081 390 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |390 (10 + 4 6 )| |--------------------| | 1/2| \-60840 + 3081 390 / B(n) d(2 n) But , B1(n) = -----------, hence n 39 1/2 n B1(n), is of the order , (39 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(39) That in floating-point is, 0.433632573 It follows that an irrationality measure for c is 1/2 2 ln(79 + 4 390 ) --------------------- 1/2 ln(79 + 4 390 ) - 2 that equals, approximately 3.306099823 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 36, : Let c be the constant, | -------- dx, | 2 | x / 1 + ---- 0 40 1/2 1/2 10 that happens to be equal to, 2 10 arctan(-----), alias, 20 0.99178947705718513712 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(81 + 4 410 ) - -------------------------------, that equals, 5.9809160998957010200 1/2 2 ln(2) - ln(81 + 4 410 ) + 2 Proof: Consider 1 / | (2 n) 2 n | x (-x + 1) E(n) = | ----------------- dx | / 2 \(n + 1) | | x | / |1 + ----| 0 \ 40 / 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) 2 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 2 20 (4 n - 3) (648 n - 972 n + 203) E(n - 1) E(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1600 (4 n - 1) (2 n - 3) (n - 1) E(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n Subject to the initial conditions E(0) = c, E(1) = 2420 c - 2400 and in Maple format E(n) = 20*(4*n-3)*(648*n^2-972*n+203)/(4*n-5)/(2*n-1)/n*E(n-1)-1600*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 2420*c-2400 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 20 (4 n - 3) (648 n - 972 n + 203) B(n - 1) B(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1600 (4 n - 1) (2 n - 3) (n - 1) B(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions B(0) = 1, B(1) = 2420 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 20*(4*n-3)*(648*n^2-972*n+203)/(4*n-5)/(2*n-1)/n*B(n-1)-1600*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2420 and 2 20 (4 n - 3) (648 n - 972 n + 203) A(n - 1) A(n) = -------------------------------------------- (4 n - 5) (2 n - 1) n 1600 (4 n - 1) (2 n - 3) (n - 1) A(n - 2) - ----------------------------------------- (4 n - 5) (2 n - 1) n subject to the intial conditions A(0) = 0, A(1) = -2400 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 20*(4*n-3)*(648*n^2-972*n+203)/(4*n-5)/(2*n-1)/n*A(n-1)-1600*(4*n-1)*(2* n-3)*(n-1)/(4*n-5)/(2*n-1)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2400 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 7976580434669599693035363333453057277474080494099416131662815031286893725407\ 448169729244370397710043224667311670429507335789337789874173151655426970\ 057506121758842723723443578670480902448185334964970560786690691938567903\ 263130624136729071347508926352060131372555370213596559460960716863588435\ 816925228821421541483619266310683153930809844462161320356638032486876725\ 935777184252049366000996464395061314427346073139916703574462495149828483\ 121213566675132541627764495050640872627178809384792847034293651249287933\ 074121296142914851746606158766513983682359036119515202595480728151857025\ 643398682416188676988403196282014557966445468709097581357579277351550219\ 573682763036338554780045739528546792817551971052488126178779796900382467\ / 28 / 80426145055879451316096593394523104336499999784148406798509634993\ / 826826781287615848508079670137707769737812294028589733711553934975979941\ 300616207407196253597934732130904484927395714492576578094661916794604722\ 996151212780023090313593927566516722182452256718310801796244316990866269\ 837705952265159209532633745858693433379541556144440805204919413432548223\ 040827754976081446996979811059705261025432160767298176993963532599284117\ 356995645683942470887233055600279173264361238319732413852913857739724430\ 998719789460910797926505482055488708283580963011703078473923419015853722\ 756758037779878400956225412299186673701943208705175989257584115624026387\ 108708447899697652346464541188206160378850126284780509884510450741073582\ 5403296164755 and its differene from c is 0.24761489296212312672839532756909672910795111761276547440083897996240596504\ 759947157275216448118531596512234825686779025982161514490357589771696973\ 997837286305498969458686669788192705963775653900963339554244002115213838\ 093931890162151421948854927571186917139925059095700261653970611265158013\ 316758380226542773848161243910202588757289362731388722925829823068366912\ 048852877862632867011395128137575261289677265299297605406229624365973871\ 739985397669113001412529861203610808143883156807937989283370180000986387\ 162436202512955226216147892830842308807458049667494292205330571968730937\ 610670350840613784930627014900523581615661727938869548275384406103527567\ 436531134760702364429460293779886339853182251082977951829767154400537654\ 086323616367805786271232601324569929460703218153931208926062473498050849\ 314822229048294900060156990982681237430772445638991174534922268222778796\ 502995318097493287152024212725945133132077417538117408540132679589705222\ 400533369507063314853987939921969945530128587352872471942583712605597107\ 705465458996809779756020899746859692490761319630498602597176248688708384\ 543670925511687264170117914439233470789278313894632843379123120503744991\ 409571509357074844660712413931356858072214983545033800310718587756757311\ 503406045011472939818375409966979483227384422222169588386600889733534122\ 280188501468738578951233846771124640940378314165788872433377176115039088\ 634729452667052293955884486491002324610468422903141970731711082539286951\ 862732596770826861597750450613838744947039105204045967658240617298472028\ 896919454364685094977269275382805551318936054178352740411918829236125527\ 676228613822562951023630883049952921592303177564348479853751437903640595\ 141811581051400035703349954197271273081628067851699596546692097335170078\ 949308257169106137285774206748599923149774652612323792906299823751931753\ 467419707030616532101433110001031307703526222443768166680575116064912996\ 962997815195986521674122974388791065028431048644021084050209107939338725\ 722930805121327609004415735241026956431265931919118256050789864426946038\ 168695862860762758339850467359036070508092563029081574734498500112496116\ 313447534860618876678645358956749221575605846947867873890002952307013836\ 562674694857381875105546685764225936562684413501526390090007396462326061\ 725527136820519015688297896066490624336671984816685558931740995088478687\ 037681942396677089420314653705352594335776703489915850693374698165283876\ 957729687697308669662971588761251972171109881627285997979500070090964973\ 661639911026971808149049212631148162807146659016388611606200775972369089\ 449681339183501059974256675058908874602869830534233572706960632431664388\ 780822798696637801685942089958213463460130791342490862146541272336778687\ 937191873728480167596001760112748273346195698731993714557413080122295985\ 685522134925636163000961369062787793142824594779706776640391779862573356\ 631888911299199721182375245844994346306842626616822476361303450768021084\ 073083118333001895590503416123189405567892599837914453813783268696220511\ 066079794949135879460677934998213255442979250338575661697894414889162300\ 387714254464247708178299228708587135457127338513373796391306364217421526\ 111054635320611473862477904066225347632337040159185020078620516022335717\ 503262703994712604919911658769323058973727377120200230777418358658884690\ 477535112862354261405923979858472929749681719924334391872469927070312547\ 055475982295995550763569123791681368378477052328339242188969738291288180\ 535948045413349016865217794684159818749135559066120921055270640525973342\ 638721882636961555557861249481173540481622186691769972243715733310077069\ 112396262447281965231119346545362379462522308308457518548262878820674853\ 560950520642388387440804599098651705911161064597190585766468996294588370\ 271986976488007925557930100041218597273803183731271984922055744947221953\ 216812290593650933447590041587193560572717490976485405941006418652853773\ 276073380079707372443027986223247859736009107651709056153597695619128517\ 151558423174168817529618517970606172351158414362373902547798935477397200\ 162934648483803876116562028965406853729359640117492834018938248763227113\ 161043362976122193057108199675467183187565259914485840937584278802967171\ 266546922106962645655299302419618700532616873090459511539312069729502201\ 119794846370293669471316651129740572581323380577028629807425115108368560\ 752453171729700145768665244287197214673931343166169111006010812716608462\ 198456550241674785012831931234859395244551436370969221384526112980388868\ 332386766624673281051930645160198722194995398033214620053733136312111276\ 066025506835082733616298491440971990288408750215200177073066926214345825\ 515612219735422931005944924175856594590065174247168954416435808801435072\ 614910096600701245271127222190114901815217981822037388783300305730782388\ 334763589919386993280481908568221910627635607948948391556493809212208880\ 043468785353887122577925262718767926451784677143857848115695163990445837\ 932188078264503104984681116010304260904707710802703190537704221034034324\ 271611442637914384783614610169304624160690982668275733225278480273120068\ 828649903605643111158124509598358667905956863736197275063574765619501193\ 716373290169117990718465416627226480652284379159655692026459576305466390\ 272273044928935341830055458891039457467136714159889131383543671405111817\ 597315340251941068667101387528408175304822228109546575696685506872959043\ 603619973513258279546430897562556100187601152768963885265705267748461862\ 303925475142603084366025305208843170546452977983467967557249497836285597\ 891615105843087532109707635412207360564796978502643991486887475705092297\ 106434486682421104762120714590247136440593886597787428145262620428974204\ 739293349283913553987043219240294099686557329180924009199707715042339767\ 169449866493848106024452294936878409083806870626526306650430998214903889\ 676982612339449853499334257299949811349297884411068350056140367278410501\ 572877199082305798099907064887782281665573760498027400640229531422956008\ 247483643199349622601678102444803239970228066874230812738435476046456031\ 311935456969634929419675754964285428078139639127119490716889609866709315\ 831398757821091513986964189552602480068429033784268480516929171072713556\ -883 1653373060465929961673314878747061571917764228485135429094993876306 10 The smallest empirical delta from, 100, to , 200, is 0.2125514012 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(2 n) B(n) d(2 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 (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 (10 + 4 6 )| |--------------------| | 1/2| \-65600 + 3240 410 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |410 (10 + 4 6 )| |--------------------| | 1/2| \-65600 + 3240 410 / B(n) d(2 n) But , B1(n) = -----------, hence n 10 1/2 n B1(n), is of the order , (10 exp(2) (10 + 4 6 )) 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 ) + 2 - ln(10) That in floating-point is, 0.200766268 It follows that an irrationality measure for c is 1/2 2 ln(81 + 4 410 ) - ------------------------------- 1/2 2 ln(2) - ln(81 + 4 410 ) + 2 that equals, approximately 5.980916416 ------------------------------------------ ----------------------------------------------- This ends this book that took, 1227.133, seconds to generate.