1 / | 1 On the Irrationality of, | -------- dx, for a from 1 to , 40 | 3 | 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 | 3 | 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 We are unable, with this method to prove the irrationality of, | ------ dx | 3 / x + 1 0 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 2 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 3 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 4 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 5 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.06064249984 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 6 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 7 ----------------------------------------------- 1 / | 1 Proposition Number, 1, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 8 1/2 3 Pi that happens to be equal to, 1/3 ln(3) + -------, alias, 9 0.97080388430077584732 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(17 + 12 2 ) --------------------, that equals, 13.417820233353765998 1/2 ln(17 + 12 2 ) - 3 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 8 (6 n - 5) (51 n - 85 n + 23) E(n - 1) E(n) = ---------------------------------------- (3 n - 4) (3 n - 2) n 64 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - --------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 88 c - 256/3 and in Maple format E(n) = 8*(6*n-5)*(51*n^2-85*n+23)/(3*n-4)/(3*n-2)/n*E(n-1)-64*(3*n-1)*(3*n-5)*( n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 88*c-256/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 8 (6 n - 5) (51 n - 85 n + 23) B(n - 1) B(n) = ---------------------------------------- (3 n - 4) (3 n - 2) n 64 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - --------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 88 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 8*(6*n-5)*(51*n^2-85*n+23)/(3*n-4)/(3*n-2)/n*B(n-1)-64*(3*n-1)*(3*n-5)*( n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 88 and 2 8 (6 n - 5) (51 n - 85 n + 23) A(n - 1) A(n) = ---------------------------------------- (3 n - 4) (3 n - 2) n 64 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - --------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -256/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 8*(6*n-5)*(51*n^2-85*n+23)/(3*n-4)/(3*n-2)/n*A(n-1)-64*(3*n-1)*(3*n-5)*( n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -256/3 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 1762330096650977689600655794405970747880694311819515086974339307944509279092\ 158817829223728334850550714861631038981578940242584537407490125585844621\ 163042843089966933683445355018931413536083309123342189644233482839213742\ 572997883027698325296394122419556691635935983866603558629600747553131608\ 778478454597743138949423647063812623403430046288707233782629664824512778\ 219266216368272223738541499431960785444158497468928793879548255322718085\ / 092906076049797814542739967233216648327966627354301 / 1815330701854680\ / 733401927710718362994352170938216338734017136451731367071976175035594111\ 548775191295274434969949206681708444539767234201372699803353963515121233\ 485618483342534834696121792487450576305782632105044119450426440456041440\ 463488056648146348311265769563932096187651510661942049052045879989483047\ 956014851569376628297357197504361223045437674725986541612526316062338142\ 266808934975568419153794126244520530875421220686413574570697809878596526\ 940130655249732693813463529541406514100 and its differene from c is 0.46837372735305927071109852811778663201976875907305314067114433369391552083\ 882703602812604043678537495934213325188824641700893528925063913578339803\ 867464612568308822438423653085977550562405948139816260777546794374909005\ 234957800098780454937620892338912897533146814197288131106911796855912888\ 155644102780275158413092350414113442146571571733990617565991278764256475\ 909373984953451082895489675432118884043364167165085646354669527912156270\ 732768801551227174154380417230373379522931446170517108755333972831920214\ 341627485677075760439739172233711518744000996523129410377758589979525522\ 422123277612725450590373180995835149950461934559260059363271627293392011\ 017184832524031788213809519153354952637551702794953687605278024779427161\ 153392236724783219242020926808898376022193221943695192047916636896689117\ 950172056217601722292684772680182710210886344251913104120262269313067644\ 775637113130204745099276800222629543632235847396164954648619870292877518\ 840462289107579771482140872339190947884601970420930298810948655083877906\ 116202307309401664158985480273137061880106852642567957059374568086083390\ 084732508925940927732509890933710897015921375685432563217985287373449019\ 819469544111137031589676393873855621224062893786835429943529838641982712\ 300215296919536408289553767353655135537107695635151977905379915684730943\ 094952962499814154184847779297455929330455113607923650840819659342282290\ 757431360144963591816683843236319716911777870150246008701914121909660153\ 953644507387942071425129634635811787031606392077242138979721150196185742\ 344441292628533597885810011026218565962720953352490377230317516762596343\ 530341879096656048000247258669470443103267548615185717421072946980430349\ 824602150121282261999667847408703595745824118785867933057598502875455196\ 852461113554411354910645225882877832816115667456878893417946576251169760\ 127391262205226817119098403449201524838704864275788365146778320919390004\ 255626662414093218798244513308586921814063667179802628971196650179948888\ 819484891225000923388236209783100304489262959612123864311874147712349367\ 561583890460515629540580065100235068412981083498258534133764508868300200\ 041977221701722170917039769750569480633390785998152271737218286710302602\ 511101373314469442184975043993819668510991684623070420583141898479721582\ 045126111792959964140463936983778964435919987864985047780716106493711234\ 452258983294061295410384831219031466541636494029621572693207973006341504\ 252741183741522639532312684089828487227660460491591049877244871793248549\ 998051640426829386312347692430021466304916955051090268139780074779724938\ 206070680128405591622608648421780386911555597513643549729204756150618793\ 471742685617327043870768894858371581551739550092924689594152399304529056\ 568130268385994224145850206513230339338353556445637211654652535242735325\ 081013848885960881235734741195959658907079189120264068515571861190020012\ 139977144325008610034094720562937075420678180146027210765895807966508783\ 090380216953557457761124094716613830341441596419520271204094404541599506\ 978099581304285943313964994218074984679053577476626478188355235416701987\ 605898112830901931693985389137044910542732420599955971641086721276718428\ 694057349090503448730887292991066986811612443477386558262729684656651568\ 370378768169773544151349676210318373413210033052338128109785032224177253\ 159670872810723885889779199730532088753674069751048808678131941177945033\ 072928254756613878626202409369682659006974831952160693591772669257798836\ 937426177154665970519138893366140000280730355615638938372244737578750935\ 361905112800769864128296448617318632522393857079679371734644221893396430\ 428372812550058719902238484745576108489362816453652409437670689655749773\ 845996372891060572734896198095380517281379850405263102092526128521791149\ 253051471420210523072191964539474451043132142829593313708954684693164517\ 240403826237758124929765533978557790725461928199995266070206618769767678\ 316234629630258674892534522149233774807845195926229173699889270264496465\ 580090965745184707556172943892578082218367421513920804724618707350888376\ 924526264259646744839482907069806115716707586115890045624634311468306342\ 695782833551755725639979104330137403280910455297250766089986744399415384\ 215413358768950778564889719961170865049200197562029159210390367406179420\ 511324819253054865760153879309164434081225219938774652533646691265521530\ 019622614715413732075070054600209082281789635715133962234259818959027089\ 982016093941478375091403358438332301803451379980014367109236337176500042\ 874495906330840470819628859420700106354761903989623619844029915657382501\ 902763987120501845180498498411560046238157640232060640776116013371461616\ 169127113990187310071785160709303394633666401084985976310667375408848521\ 563598233878122025570901386137093540094216722633908610243997380801084869\ 587318160490500735077823669446461055814525726199464323829141688157493023\ 344001814948428473315207711147691914315052497331758724661296740461749572\ 383788361332680044020034682919986286151756159768757584191163134640816865\ 680536178361025204153622810992648644567289346358480814267115771493102347\ 760845890087592715551325989461953454574253224112670545762244121597065025\ 381947896157642069863569599454753121254561981901402652429808055981988330\ 689222508419877932654212320273357845205091081245747971787359779467814221\ 959916259796587637889139482275189127505084582159388504660729725222537936\ 168838440507814555249764364538403471463089830384862046638842808888567376\ 770789558957311152422977849422742563917004705929175998458292880834048076\ 930626726857646225620810619778772115635735393941359076785072019182869604\ 319568737973800241964970430227960499745128934335285073654992880070362276\ 355158647583085365386983858021941633541922430663366680524344024361152434\ 052916691245638127349820138375997127551420857529759674679442503630020560\ 316993519833380529314074450108464108326930262064522560247512091500317199\ 783376936476822922154653672680852640454223767966006049805675814816986922\ 994347986920633600515054321134025989746259022513620329544353369151628564\ 396127927875641793484658255065023312085783553674971442525985862257835801\ 818813683632915184379866914353893551687085751361956939411341213385545591\ 420699630307139511389712217377617926244532613480941691417861476067028903\ 900717834316493263123299310816522593845129948023355138678449735175308604\ 409340729890816099928440033284220086963908632221395398399282981918542329\ 524193128319009548254353978686647621391726332918950969918928120870209660\ -612 09345301973387238892314257712355136694818860062766 10 The smallest empirical delta from, 100, to , 200, is 0.2266147357 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(3 n) B(n) d(3 n) Lemma: , -----------, and , -----------, are always integers n n 8 8 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (136 + 96 2 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ---------------------- | B(n) | / 1/2 1/2 \n |2 (21 + 12 3 )| |-------------------| | 1/2 | \ -192 + 136 2 / Hence | A1(n) | C | c - ----- | <= ---------------------- | B1(n) | / 1/2 1/2 \n |2 (21 + 12 3 )| |-------------------| | 1/2 | \ -192 + 136 2 / B(n) d(3 n) But , B1(n) = -----------, hence n 8 1/2 n B1(n), is of the order , (8 exp(3) (21 + 12 3 )) 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 ) + 3 - 3 ln(2) That in floating-point is, 0.080529469 It follows that an irrationality measure for c is 1/2 2 ln(17 + 12 2 ) -------------------- 1/2 ln(17 + 12 2 ) - 3 that equals, approximately 13.41781440 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 9 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx | 3 | x / 1 + ---- 0 10 ----------------------------------------------- 1 / | 1 Proposition Number, 2, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 11 1/2 2/3 (1/3) 1/2 3 11 that happens to be equal to, -1/3 11 3 arctan(------------) (2/3) 11 - 22 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 11 ln(1 + 11 ) - 1/6 11 ln(1 + 11 - 11 ), alias, 0.97838312675996323166 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(23 + 4 33 ) - -------------------------------, that equals, 27.455615912910284016 1/2 -2 ln(23 + 4 33 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (69 n - 115 n + 31) E(n - 1) E(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 121 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 165 c - 484/3 and in Maple format E(n) = 11*(6*n-5)*(69*n^2-115*n+31)/(3*n-4)/(3*n-2)/n*E(n-1)-121*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 165*c-484/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 11 (6 n - 5) (69 n - 115 n + 31) B(n - 1) B(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 121 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 165 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 11*(6*n-5)*(69*n^2-115*n+31)/(3*n-4)/(3*n-2)/n*B(n-1)-121*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 165 and 2 11 (6 n - 5) (69 n - 115 n + 31) A(n - 1) A(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 121 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -484/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 11*(6*n-5)*(69*n^2-115*n+31)/(3*n-4)/(3*n-2)/n*A(n-1)-121*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -484/3 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 5574451003552161308571841425311709344993108908205448647225315648832095617793\ 966276987492476702899015578372701372947278230756520716675543777593828980\ 047820275855830190792053991990960742809777951546747915193413614794838061\ 706045986175389054101719110546157455267601026717379530899062672313853664\ 267980873495507866240852174293866397784648308983427261774540452634776283\ 918292598240110207944791659896633334552073151648529277646854609829900674\ 481768004639186151571600025324801822040794626187633349398134164384275881\ / 10967180716979796372255157309243716372893765243023391613 / 56976156385\ / 818360025901523293722200949077970681535251333725017379232948843186024801\ 184941409106415212005693768305025630210538283329565317404080517660731702\ 941319309714465435955998113112805825435343257973746630820747601714705340\ 648623248797420419446552220716300198609203470755870650250587473658121990\ 100293998525045874978113998649601216645232464115255329925162869643372003\ 930427649346011457255182967491037798468019824927001716555874450617842310\ 611832481968370422457618439963335321350430237672540347064696650784344718\ 6549452058577539671228632339558708510473572243200 and its differene from c is 0.12387648104359435160112396611307681447399885225901132604340550363478809983\ 056656973974951423372905893835837552302062071146294520018848866739439364\ 315925056773019937973417259791635290771572032059242101105011719660334563\ 423909088376693243841927967495102636603450268682033655535323261961651438\ 767099505255665240626855578407784348969085210685450805863875518459577054\ 245068206992300204778904796526319227147189437773296858834875679842431613\ 123864891571473084946785473843834431599089288282113621823809191515346326\ 476276208129980602378430887398890947199600015122106250937568550663458539\ 473752944668920016019542011200329937533629008370932129489538664587011708\ 933858506430251658145825672186230284124158957704209778608110535974710094\ 327070830683185776808995718651267183964233804034740097411642554116986862\ 149516000837969098038694061952799988284782426340059830676887810401394217\ 415737079991820373262046118005472671545780403962634960495336018758590001\ 411368520179249512419035872152087611500610153103403943005366313621464237\ 327654425070676202008895915885161128507314400902853001199682975816710871\ 609586974736419207628250271711855392953905203904429349443631261900920431\ 568935831978990448450858109172445320623637066088763614998007477514611554\ 466658714436887912199058896185082466803540248417080269760045819307778360\ 000176483255091077121113575984324777840305353642306243244376318531265306\ 892073975765072030068243590848767813591164971742555957236795026840119132\ 901807042171062798792943172660134778419407460372320383790570427086130682\ 765896289455784233836406626201162048072950037866233232088462617589642690\ 550189327427752266731601947871524981188691864927861973054240015679321277\ 526483845448884335546948438841841823605370889202300699760498173125444869\ 577951518135382244920320779880788229781926065477504953315019783196413548\ 870776667123029437097932273131445550663494609122658748300599918867012589\ 134350031937379443671246135301105243186795500245275220098885739078279524\ 951561383452374697072554576212145133781568855026672432669969300169447074\ 342449019638322952815396602741323979270604680145590277629700728753880947\ 970231454574997047674901667726644004175497649039269020797572488429458370\ 999283451187221673507673569318419821724809802152413700468964224372170487\ 628442361361475257245354454611553867413328753992194470126221151923407192\ 225109243146862185782153616673870680359754266861973926734750712302309412\ 808744592561468936555436592728244042872201231020532670921750742527918309\ 392981713141022014227597347865461879102640147139140411429123489556896143\ 624989731245852747737781776357929450115529094011942271599877041502123018\ 524246293933411157536915572669422806508284921655189797049146396175520549\ 247976215293985640484923122742574126544921310886066580259746906603754675\ 352946929848265092138648419496758372345350136301816045792230323403072614\ 338522730019647797942808151250578421482423542462024498123240350048005930\ 564497276701377479165043271220566439354689802347202742283330779897065689\ 170233268599301748641193017633116996638349947929007230403445755159473214\ 974955008916837516283560302593051682701260455560369961020784893101259791\ 433954590445126065923308623436066461888268087043610792772574340946856539\ 651522547140546835174748590248502815098910190530977094754931127060173924\ 747672421352178582875832875845303449472548703074577598694056450620512731\ 352555612205628569059191677491740255570465602354942136537582518038583881\ 184194579544278149959885184279432019524537925081052333699981615755785175\ 497996240925269032391992418546381436002342351479465030130727769701125185\ 852493881122981078252371327771417004293013141065354345892792293340195636\ 913866167648353163854361573239568466398456718193830254790846901093550371\ 623772278116028458460020261650741710356915620443782134625005826147231457\ 602672407346245116426113716374937115031390347026137493230817079430141471\ 289547331089970568259069254109895261461102035547229157376067426502725309\ 977822774756864006744575525526006040260519043189712533238168039402457633\ 552759434514412668541079271648843462011417349473956573723030438477679740\ 252106084373105960838310783530456542811091789349090121408731903361250321\ 872025269854732614406771303649634824835236719506740918358358603787875218\ 120973660847646111607766669066499193576932993810212013037807550836209620\ 591720594940542459243675544444138741649406737610168129559800023386972597\ 155659893473981754119658352688911999378060084052890803793046883566252538\ 632524919978913028631078231232487658089892516889012583402005277553076511\ 392311308598410973768145028969831541792663840834693994612626514330160662\ 253332956783474761010360355289601686944337833714791837864775015860410301\ 007881454123187177716751274974491290726977769298668880569470801970968283\ 252306252447525657112073238288613001162311395020375667947504734479696764\ 220217541666592819321042910799846093350824930994155751128805843905679208\ 845927027066978682822462017957507192997955901741846250261517866996347345\ 667293762679331696465172895402912597895347353111775521426522043592998107\ 927496881738402085306732415570005726279428786454521108992987331314989286\ 108722723549197989638239800350201489490947916225942674143523154574942314\ 098296430742104517088393047131818969454154908524557942912699558268158777\ 387439092900379175020839038926559083457123926337750426577455861950686749\ 977106665799238775132499347723006168205343424071634929172391627270954181\ 283394289097688676138002770297963213998672982598752515285066079296115883\ 451508712992667326808702990694366762716640987409289552933027464307143913\ 065453006280268322635459169521709251520740813695155463566894904316207473\ 606236021441201155229460178867099582636644299644260701094562848109099602\ 327011663654048138395438275501760374559421240878222950229760196788744789\ 395329540217692673047222049948510761713070044453257371704794626872887568\ 108389678450643227111559491948999293292078588914418062140676955472378449\ 038377450293192051140774511307240850259385911879047798459954920589232912\ 131022527177766687652629621926366396438546489745613493650418529389173275\ 243988447624059010123060146852330490164765226566453402171258872198899006\ 484333909370620906817343670793835492600776557445404127576414619116465658\ 650210307376180898768204873847300444740005177099149137189684000850552338\ 474476470705605425020713361181272180633525255856911983072037065573027916\ 3829683658753940687736283606725213137662326688164724137987100375322966 -664 10 The smallest empirical delta from, 100, to , 200, is 0.1707535118 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 11 11 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |--------------------| | 1/2 | \ -1452 + 253 33 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |33 (21 + 12 3 )| |--------------------| | 1/2 | \ -1452 + 253 33 / trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 11 1/2 1/2 n B1(n), is of the order , (11/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 11 3 ln(253 + 44 33 ) + 3 - ln(-------) 3 That in floating-point is, 0.037799116 It follows that an irrationality measure for c is 1/2 4 ln(23 + 4 33 ) - ------------------------------- 1/2 -2 ln(23 + 4 33 ) + 6 + ln(3) that equals, approximately 27.45564515 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 12 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.01338998544 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 13 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.02357318553 ----------------------------------------------- 1 / | 1 Proposition Number, 3, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 14 1/2 2/3 (1/3) 1/2 3 14 that happens to be equal to, -1/3 14 3 arctan(------------) (2/3) 14 - 28 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 14 ln(1 + 14 - 14 ) + 1/3 14 ln(1 + 14 ), alias, 0.98283716971401122592 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(29 + 2 210 ) - --------------------------------, that equals, 15.895974621346866328 1/2 -2 ln(29 + 2 210 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 14 (6 n - 5) (87 n - 145 n + 39) E(n - 1) E(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 196 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 266 c - 784/3 and in Maple format E(n) = 14*(6*n-5)*(87*n^2-145*n+39)/(3*n-4)/(3*n-2)/n*E(n-1)-196*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 266*c-784/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 14 (6 n - 5) (87 n - 145 n + 39) B(n - 1) B(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 196 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 266 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 14*(6*n-5)*(87*n^2-145*n+39)/(3*n-4)/(3*n-2)/n*B(n-1)-196*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 266 and 2 14 (6 n - 5) (87 n - 145 n + 39) A(n - 1) A(n) = ------------------------------------------ (3 n - 4) (3 n - 2) n 196 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -784/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 14*(6*n-5)*(87*n^2-145*n+39)/(3*n-4)/(3*n-2)/n*A(n-1)-196*(3*n-1)*(3*n-5 )*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -784/3 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 5601331570877965736659582775220958503350857043721102184174585141219744585308\ 432165814708778882607019523148210731543104666015942348363669427278881976\ 317482686967441912476709028984734541641074196539700562555663737379148426\ 546173177277121582717830763190044431612183657315968703470145554665419368\ 803051264207907812006999814380426329120884195527356347454854394172010856\ 448469497594522386236068792337265594167630335242355204330705541279213641\ 892446996868542169109743382159401339946923516903595470303430551415049037\ / 2843436262495862791499962563352938042565045346560834914738658535857 / / 569914502979965362589898900631138182234091556397536770015123483895241600\ 044993967154029022701947463899914188379829704731286196630247386323559136\ 722304532708360629256933077907198180764911523073292886483457645574716731\ 543893389565289253306426886483393785870957570120177900865049527660598764\ 257669593032380968461566635872549494687194530163660553147031735664741593\ 484127130216500350192197794809052327935435396530419644815904028421079429\ 830741889153657701871709179611856783404911500643961067184174434762533898\ 50094125852391989728081512440791689007884507673558651266700731616273800 and its differene from c is 0.62482328937430960150699628738241315917711994485318899025634230841650721095\ 842711797543938911322260167429133681456040731799400332480041939540757309\ 960802192999454015033415189975177201522160706275245846457678771701868525\ 070782883975392726484118058553597907301995637809906245042694858572336469\ 506519820764103791789086140584865799856900120901004700849067738258003175\ 498546296195714714616808860509783359793054846875787151497568713570602961\ 703687271639484061350345115471398831957003712691066873972492130669600338\ 809040790473429986664689410547544001503528229564629820258212645458863874\ 770053246515247777757052411800451419290699485238813207687229381158096220\ 141422490055958711262570972847826325637686847981614423956539247464158375\ 057208918159116434516543802040300019728157926055952908965802125735090753\ 333488822246018482107103462645894080205014445534252155586306706086669941\ 122497744708492575732176605125530158463353872437935318418087763641533854\ 642036414483491904076570950270995979743531686053017756733681539270563104\ 939794273964941514188471720605916194309251207735484001833214531407399323\ 559271117230627958190036597298569744316691547475984715306863079161148509\ 036514646684070591080808606102935874217691673533720968500754038418869259\ 846689083492451786479530216595384378211286044075942006944394709940787059\ 595165352538811000247729652366318223568118318441656027655232119360086888\ 427852889916647779222238600233907382767689674695910811550414700788031085\ 736536023768612033656690490235917261656985387760756733927101387859698243\ 833956024036490638612963312521202672658500421767083066552738649625203485\ 521159385791397269495154139501414311907634479938340740584751913468536386\ 048974987269942911075087756280428764961455983868034745119803428050964928\ 376452934515544670340515584336477707381462481246955443220370514327139223\ 127747277673760886722562638861729490631623168987478906074592887793109772\ 859833893950775749635590293654320682070924736951898320326680677672993970\ 734772259464334264370191824668148388398420664440066730406895578593040019\ 027283844496341839345356519861386320459542690001746550092698488275788374\ 616455981170215930497897580014083684442280225925199118976955435591855994\ 296911900187860762528011457275173950480973317807990365441491168445732498\ 043928496429901457462892084887208908166767023696902284104324318852269126\ 505763946972759154230843650878720282966692784881509043558637386131889325\ 936128294336712510530880470705702315691291622001113968611714325790067871\ 128996348432988356563846030159685376009270438126336264162405641089422140\ 656734082108739413108256409658859618996793689190774780075455519311186267\ 091011001176346920558060974551221561993449838356207978299550724069009115\ 557128465310510068262071670670294320482187379667532091763932294751427612\ 877140155799995060244372018180445784662451393603620120952738845613885273\ 631340713239540774333279421304499604239923611502252953454487074997242633\ 384448052211058132501167436779956842483552073882110288662637012134678581\ 054651252789737358192295353942361181767029199919430580898225885856847968\ 013669388096727215551681241755339002517912457566735326111480624961869127\ 175867498739240034420229248775588036733659876844657201636332667091058739\ 582538579182049349889290771254398044526901554901044872759446986317574950\ 910175995866380871012978293280756785534344326626761350742117332201816548\ 624844325394812286116619899407443055018312529148976166360179194019874853\ 774983632740892313378357787222545013864491444415485588100416277310815940\ 295563038382239009415804772553060070133645014863481597578351565511684781\ 152988486039254918576987803290613707253966757657224834096787346650062182\ 052506462334354684001343080191966987095628061506801346589504899461227927\ 531872976054972757819058378772460537270138921114371545721808456584921880\ 556290528380256824420641226942243719236647692082419424727494516634063097\ 015165614959321498263689500261245232342692673926627991545586301180546288\ 466889664620592869547233496773995865634229421819728416134367501953544867\ 121564346090705800306421410258390118628216351381368084495206553539062241\ 329042995286875700077715762416938336377438312662285554642235553808431185\ 846481796348988253819628494238494776411498126120651710682221195071993750\ 389413759063767032857078462132638320102938895301819807254460131142157659\ 099309995471652675802367933398318379721364542733440327824266928324167679\ 307696557117435407603219994036092336272759524738657620002299483556960369\ 807497267513460654249205801817518584747204856629876520243504228914067441\ 863395813581063175567045270001858322230491866006299326534381495359813662\ 426810713169837406435541932888204960061748462289272776451050683396359289\ 047922312627517664548202765033898569064006237413035902616582794407713968\ 919813565324776960470486599881738691758424610429022625755403526771299041\ 603807487167151375073018827030422069095929933795415955156923602798492212\ 407193483408889847396187116716150147516938409978799631621689280000408852\ 888817999802366014921596084426748395117498784074893476881832654561458117\ 074338620295600567385008012415020127816149024423592961864576689246244615\ 741519548217112801717118031165532626228619101128723587348076107978855774\ 529061016124542511086659297249602078507942955104731333022901595201361854\ 504963273390733319679743614008414307381147257329902992377386951604898361\ 773582345524898045389745710780035894204081362806358367989128934671700394\ 642604707535652376635824048430862106218192093848554165559708100320718650\ 599343827867338516996663628185611688860979456470690295011490208189196920\ 982660148986102294383063894201581812836155196324504754801706239189362671\ 229562930880212686909573961145841218974919071509610023859984003291796205\ 690723271905818823346945332315027242666963656786178179672613800401898534\ 510266647490264109250752797471360003639185902189226084804783977780381510\ 104434918898778578300515336102105788580487305302917902198169103829710627\ 395137977309526302278229668824188764996075651200686186841674573068923169\ 145406336293783817481593531676693047975515017791981490297459682184360008\ 040804234188804919858641960477741463520930759219413678931649769537184224\ 007664489571641549439713263046874142295788218907544206334020447815584778\ 376469442901009421659890239687144358737553399586168493403226080401834478\ 964184103329399369548750516800471616170787315885234869220101363305125620\ -705 31502103831845638303003072675 10 The smallest empirical delta from, 100, to , 200, is 0.1987758041 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 14 14 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 210 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 14 (-14 + 210 ) (-15 + 210 )/ trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 14 1/2 1/2 n B1(n), is of the order , (14/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 14 3 ln(406 + 28 210 ) + 3 - ln(-------) 3 That in floating-point is, 0.067132230 It follows that an irrationality measure for c is 1/2 4 ln(29 + 2 210 ) - -------------------------------- 1/2 -2 ln(29 + 2 210 ) + 6 + ln(3) that equals, approximately 15.89597471 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 15 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.03947161057 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 16 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.04779717054 ----------------------------------------------- 1 / | 1 Proposition Number, 4, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 17 1/2 2/3 (1/3) 1/2 3 17 that happens to be equal to, -1/3 17 3 arctan(------------) (2/3) 17 - 34 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 17 ln(1 + 17 ) - 1/6 17 ln(1 + 17 - 17 ), alias, 0.98576895791730682937 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(35 + 6 34 ) --------------------, that equals, 6.8065711675943530250 1/2 ln(35 + 6 34 ) - 3 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 17 (6 n - 5) (105 n - 175 n + 47) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 289 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 391 c - 1156/3 and in Maple format E(n) = 17*(6*n-5)*(105*n^2-175*n+47)/(3*n-4)/(3*n-2)/n*E(n-1)-289*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 391*c-1156/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 17 (6 n - 5) (105 n - 175 n + 47) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 289 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 391 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 17*(6*n-5)*(105*n^2-175*n+47)/(3*n-4)/(3*n-2)/n*B(n-1)-289*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 391 and 2 17 (6 n - 5) (105 n - 175 n + 47) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 289 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -1156/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 17*(6*n-5)*(105*n^2-175*n+47)/(3*n-4)/(3*n-2)/n*A(n-1)-289*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1156/3 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 7737067847319357723828735880243908328814524208684422853404817603986932046423\ 581355643677463687570408821275612874677343798098973591218247189788603166\ 310586757703751497675373656621273321951661314793652394863552934620097833\ 870203243952995808388674914460399110563189835241665459451302442608236511\ 277868103487833699354059899522021749531115506629655033403701746257636907\ 736123907498105083418650533223073219230408711534513296296673672034675397\ 350077871605305077376515745713049327690241999082458032362723213408705093\ / 152226860764602700667628719182095714235850867 / 7848763937206873275349\ / 364069783014061115778779937128510232548535444313196981889378141893043628\ 548479125778121302658889837312373974096176996019857712649364633891545338\ 879380444380291516379632476258474221427980934623075390381332887092491263\ 907204309454909064423038904707018750729611729034154433489890370980608469\ 376536450711019571062585072125924372187663945838556514780163129673239080\ 395178615035485769449577250633064797743133935146095973614263172763614236\ 238426554497557775292832164247608831202755799068323021195475559673658926\ 888702931501594480286739200 and its differene from c is 0.12954065178690338441652144160497987142556019439297019769650792920464392816\ 652104863303324590334954068321373376182231445929530748634874710035624077\ 037484072614716784140683014375119976801118254127013660220438753397996053\ 934475059408479818244546159777789457902967617304068976251138652133759475\ 763798390629145722505619859499801847448782503119446578061898561015309995\ 021474640480277209166100184180328610526833027531106504757648602090977864\ 171662837605140925340923014299126683615824735762576239456979554967804143\ 599632909143356088687651007998440213372939730880672001127907514636422904\ 308456552337474942009104824216282029632686966824204450956895340381227982\ 283142937841663907304363564837755635677287999068408982931061446792902961\ 306451515907450904207587092073328469405947419317877613253586407378156613\ 409302503121013871881620222637625923550525941469733609786039982300103663\ 853493665526962990098948662893466445191229084455140641083396861320092834\ 745258271262952217346601269630220723263346290762368589724137598009868332\ 698263333340941503293902764690355085133886923455991611357438942250501056\ 477892325191702507456190294367496112999164169596404200104543652242921308\ 982118861852529256234596147305871796220178115485248684734340539238479255\ 181946940200471390350132636335125439159074184580620288948367053587645589\ 551970814308561256835064561121958560503659000148497446947453387116476817\ 918323412110568228254817221769672824879908907398104486375222657412435154\ 466157326603452967638492211596425922151817622007977045540337615305407064\ 122280580438814328981635135625159365322817693785475788538720937699928156\ 108150037207714479947820367827464463917901441401205478219812718277440648\ 463328459564400932534761581967550632932524946041004911194394334438014880\ 953919890196968392352087476475824351973509130614655228121719487695502986\ 985838390803167597320320215613846856684036997359511641686544014509833405\ 885938872519979805209912931207485873374666210420361532687691902483016465\ 372496203694817373642146932259958431970728919504224890579164885266824931\ 795780604033920147171737830701139587220411813319698267922885895142771506\ 501592268146380708863100455296379060525933650974205417821992677509122304\ 559978762868586011445056444743053624287139802473975121784172799903154948\ 802264170686166596640492968077303593695779014877659734205706025346235879\ 283162832234313893798241208267580575023931464832309514602727030563114363\ 521897325303389165046022146778261505534341801667353406242458490059514284\ 970440343277626109810566245603468593037988002319288818676330444009500409\ 179908951429546174168772580174709264801988704396049798116811667453137062\ 639348377129416347787387568240341704375218756941527407870240615918192611\ 365692381931761820366757709563097947781960893752286500834218911377549553\ 403279928983887047881981347744656536477951180190872430126229042445674933\ 411513455197465823763692473644215107269028377867840583336663474350190461\ 812472504536235059146923518607235952014662503402563833726311536805787931\ 377602911702498585358691295047366148954485256839950016430127310977827488\ 352552972455258064042969952195856821271850005268388792324152384182827617\ 866405816965073836609592134784335625035074820285737331253386711300139661\ 603873530672807890302621840696954117553906686353581950875256649285205494\ 578234559976583371324348554131839392451556746148765709423418694139229068\ 553101964567080734424712097969462957525409084019185456096556019979382083\ 579722852381150413089859603134677993508522513639808199859083325881431374\ 179165283281813470538177699565646970458814139926947255250124661311749609\ 920910297273757843431439647264380023180824605175778407945297442976326753\ 233367651856619101918387468737812571758768910501974713848880940977121448\ 681610552142679030542634707307202187896194276810798887586631000040447267\ 449567862827568604804625066960414045976366919446765976746110905452925435\ 191740360445866459035702582172800790743057933532288366319662283501615275\ 902325006286530982975858314771645101649205055838969744602557931747775755\ 121627679893453383696002310130540197123428281258920390106976055422182630\ 291373028545648259727117846303736600096142850090933267230676893290977595\ 050667468254512340395968202207443800996098504848616726041347084085866237\ 734413710297044578339103920656926408116504308267402349495729687179090117\ 442233344515268268199202552391556856856662103962276505900842640961248509\ 378219171670689857147368754559770303452109286802566980146668127361894753\ 430963672736708913604931034602429183843142225248679214508068546116158406\ 912547784023513326660038683606134175782766812632397482449451613570512798\ 653123033414312573705845542050953189196836274878987801992791439589460056\ 202979958702815400304349433236091544019051488355033679310551338152625897\ 499153052045098367024163166912870943921659173509899442257859775058923655\ 012480919711593646352588934568820987312482208669834440524646306592980875\ 182459319080152625289648187205764142089346284621876188085450947343597941\ 027378512054911565789238642484939292588685817419516677132280867270519872\ 653472996950581638947833835936631264604998456044498287400817759800305901\ 560001747216592640297763767770720789235411665496046131623050380360457026\ 986926466971849738872944709692420951780088806960992673619438578070734639\ 541551478246382978143992047977909901213583852729216643716198739178884315\ 447980527595260362625896215269365318081110375851764473072937907010987416\ 095540306648300608663161609445033240004030726752074065971788291016617216\ 934970685963705907105091280461568616465045132866340863369623597819298251\ 414307295159219026492346753149118814729173152726833711178597666118565353\ 672174722281269209713384032636492723787098072228813439093315706699711479\ 187972682847254775293955373000684741313115499714464764368012976616719887\ 533088241037673643520442368622890437911801405625923723123283443948984746\ 413602002030401262468392977791209563862268846273297305006275963563615057\ 040119317695882940872630923289513739746405891100782699766131066833682554\ 111249531470798113002702039888193044433492646722400967034868194466521570\ 806443061075432253775933403504972096088827219867451048348898862992323378\ 871551614520703288151711878003486882691354499013684296583460063642346165\ 259691530763978563840953167674736663292979775777061981516902050194006152\ 845600048295869485333130089581050895565274684996824597096270346208073 -737 10 The smallest empirical delta from, 100, to , 200, is 0.3120859190 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(3 n) B(n) d(3 n) Lemma: , -----------, and , -----------, are always integers n n 17 17 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (595 + 102 34 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |34 (21 + 12 3 )| |--------------------| | 1/2 | \ -3468 + 595 34 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |34 (21 + 12 3 )| |--------------------| | 1/2 | \ -3468 + 595 34 / B(n) d(3 n) But , B1(n) = -----------, hence n 17 1/2 n B1(n), is of the order , (17 exp(3) (21 + 12 3 )) 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 ) + 3 - ln(17) That in floating-point is, 0.172218621 It follows that an irrationality measure for c is 1/2 2 ln(35 + 6 34 ) -------------------- 1/2 ln(35 + 6 34 ) - 3 that equals, approximately 6.806573030 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 18 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.05654292695 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 19 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.06509013733 ----------------------------------------------- 1 / | 1 Proposition Number, 5, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 20 that happens to be equal to, 1/2 1/3 2/3 (2/3) (1/3) 1/2 3 2 5 -1/3 2 5 3 arctan(------------------) (1/3) (2/3) 2 5 - 20 (2/3) (1/3) (2/3) (1/3) - 1/6 2 5 ln(1 + 20 - 20 ) (2/3) (1/3) (1/3) + 1/3 2 5 ln(1 + 20 ), alias, 0.98784510488347644226 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(41 + 4 105 ) - --------------------------------, that equals, 10.280540651290824596 1/2 -2 ln(41 + 4 105 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 20 (6 n - 5) (123 n - 205 n + 55) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 400 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 540 c - 1600/3 and in Maple format E(n) = 20*(6*n-5)*(123*n^2-205*n+55)/(3*n-4)/(3*n-2)/n*E(n-1)-400*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 540*c-1600/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 20 (6 n - 5) (123 n - 205 n + 55) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 400 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 540 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 20*(6*n-5)*(123*n^2-205*n+55)/(3*n-4)/(3*n-2)/n*B(n-1)-400*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 540 and 2 20 (6 n - 5) (123 n - 205 n + 55) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 400 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -1600/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 20*(6*n-5)*(123*n^2-205*n+55)/(3*n-4)/(3*n-2)/n*A(n-1)-400*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1600/3 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 1020717716901927210033113112852194603982449144604193206394599826107720999501\ 077233403752909115211039625714782098360945447898071473072681849948011056\ 044266286917052189754363560363059859892084024601753529713375482004121878\ 608379120556807975218701928911091470358556296929570548079522261991896286\ 598683306478110759027694952810295733731648307947054642229321788857098478\ 392198809331229507703938313623385381827638196232367575130813054832267739\ 338001853482593374583468649856115567124220840996553692197246486837636842\ 843037427091609013884935356590733998762480126645199470667795566414886976\ / 929648015480288950600436444575 / 1033277091576344180917137031412868231\ / 407003812904755323686225603606023114183475165283829302144477485516560156\ 931900076911426620722971241410175131008062885763223884608113151216239351\ 790195554770917967597185583372562267358194578816746702461457273277967288\ 720036069695681154865476467995398709953765391129329839545667255642454187\ 500405180566804943355222113710722693982147158579878611132992533224581593\ 432381780003343416815607774239856151154762949280058291821756577627868650\ 328196205982566044968259502028485228112273374687544276196807507415404227\ 586929150568514135576329531514963441153162603984741808295927064457976 and its differene from c is 0.41406701419625420714285292231466571267422687054572360843509803483934029363\ 308679098545847903341762379727172706374283409710520306669743419822008373\ 123180153726892012765171249674623984535399757857818779252176308618999074\ 592180502782974442173940443768796444062448120986136606941263171293469861\ 263788816303672991135829912466993433953362150649802926805898571163773212\ 811576058239557802923093642704249405582512017631053358373195553557710998\ 992208938610790869544607720254354227618877819329623254820978682514364773\ 247554226102204182867427082801414749038577629094430763330436213182877431\ 263424494824332414471655530885818478906099805548585727883025906939643392\ 955588584748137627369345940968425509844950327255637907948065168779819551\ 474747527714725430866202412395592384909998912180350765081526905100363286\ 743795167959606570296749378542652795300243174661778618086709413087146139\ 366612609770732750260023474647624804857912372297638742913005515289222876\ 449001918530865159369199375041179173737600830762713591840225512882820812\ 062681601225265816799377778044661016700269471111316065081679643565996264\ 356530581639544170993479728016693714826978332550776560563075762771525027\ 768145464519396546348646116380694361363461250009533120847346948863415916\ 904230949609523441780859601636044918581628611593818540947552900527251288\ 266321284484514445814053042690123854253348135752107294324075554811998535\ 886290231536842317624804578017445155256998386067986003570070881376654094\ 912229222179754489826178516338389070739286141705560872136745432796064077\ 587376074728443414365864784821486917894461103021313373068133317610099203\ 757016974546762670339426394293721688703260700125896722140906857236443136\ 604182482230406488668077357934762233090410088691981060389201955563533971\ 967561708584200125974787195192189451678099406976553898615493943065721598\ 562092872862968929664215098378379907359253979875079220716194837908871448\ 918790708020715167915059947280219132934745287526983318743702352588432026\ 394186722075225212610135357245870641074186691053354355621354619222296160\ 257504837123559167242849861519636151250056747104437335212546669179243266\ 270574268515076126628153996285671863303762103235582095068972252459488971\ 075151892685982339307419313430817649917374966614974777710522799575225532\ 633444272952215260919135324735887505641318366201500126321612144391658582\ 156720523756887794540760417118419186381161448569731506989884447984917095\ 481002887189696062452780429880691758538961718692876658754088013998882018\ 627209960957561709733972426477706114118106853210817539694862967416838349\ 873765667548644199449235774511419937241405682152288047382366304254632259\ 160969694402212161842845914108099617371915970549625168949889445228735806\ 633544888758988122058145669064404755005969985419900571704013091364284927\ 641328255876268979139558717707686919428327936797897242078141230225651728\ 026281857470682216046600144179003880825891565698904732613830620486517720\ 342158198315027851078969096703578067630551721865224547502938693003476812\ 518135237919802446381443455958928204365288274319467863787319128069494741\ 474848701001082705055622457914378072907352344969135178056893820637684830\ 538200985033204401168446827466275287715886410971847952133007807150819161\ 613599409448859834557855758997403262142063147061926782174354107461090332\ 247176189925229439148277073460382691254464472420206516688555801937925417\ 519539470575700799540885006448153326311700791698783205573773456353082595\ 423998080437183326343837139402963099298353050338726023021637041497825058\ 150880059390942301204736244211772968602011628784127189110193968920036384\ 760794763489906545263350311832233917811092078906335351731929676678659330\ 489933205725483613210966354767466358421299520661277918228153760994964719\ 712607808010362567852483875625826232750633884494881179847496451662648643\ 848723920866281249447268876987160362925962926737845475953863929326939550\ 143026432366292638970596745975616037497922038364789517567338639186455225\ 956168602211738179350535966836502651760694191574686451950057780339195759\ 562252161011592546474600042966401445470329790149265414562051167762198609\ 448460875946612499284541109880725343167159869452280536201563629994278379\ 866977034477323017341439139782639136706351370112480851892512260906918056\ 795345027975396383859370272253571002136884320723558846779628454071480341\ 639846377282532474601455801328117894817624124947903118616201025630448252\ 368167096748663392164762766958399861187443920688729164434580266930112152\ 338678917492272687550369968198800552537355177864594958299449838373689879\ 420555396718140026954192883271193725554693482956333491006093222045295947\ 948245149300803775120488811017705920249954642128899565995829808565732550\ 622308750654108329811908376018111590260208230228302835600635401486888692\ 140634963595173212681862794035453148794355777260861866547758196346732413\ 873022935140107097320238612365699735746374394646045384670425091436789791\ 574522816947427409368331530930996246613633353930957807670250494383273821\ 768597424119867802547780151982044621558189986108008263091927884355137616\ 958435782038764639679978506019253709276271386513611255040900836412871001\ 959443207601072483107643267498103957223594267686701136021575683439100896\ 497831264059840624071428679004961706116952414768986772699499468148740874\ 133620958744796832256777458747217251720922896842708057109280812996564399\ 913271708172791577761324192109792550182269519360329875626614640166866621\ 114945167173070823832761067780728723049177428207819137654814424955897504\ 288436196417807738874383142977969594708698571093689748533745205750549739\ 657442790918049641300535554033612492474575243912702005295349475209044817\ 215552324729096552053608594289300289772276051621419937384730860716129103\ 704539671212820507425505875241468986842983220272639979146229482188238849\ 236384856781267995857971574777673246204578068236594582734512936669159751\ 658415503182630248260993319128301314374599718914226059215124613240793865\ 397370476598715488035575851332902700177838850874729314592642508042116313\ 555456074652524884133414604614515425940556110010542258447560038898932852\ 695245493086131625273573911804347533401286859082100652320736641760842975\ 399432047908341627889848566182897423880954494658935960225552187899485082\ 744848473521962496038520694654380491075788770240675908346851517399167010\ -765 57678734795743958005132277013534625563633 10 The smallest empirical delta from, 100, to , 200, is 0.2401824095 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 20 20 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |---------------------| | 1/2 | \ -8400 + 820 105 / Hence | A1(n) | C | c - ----- | <= ------------------------ | B1(n) | / 1/2 1/2 \n |105 (21 + 12 3 )| |---------------------| | 1/2 | \ -8400 + 820 105 / trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 20 1/2 1/2 n B1(n), is of the order , (20/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 20 3 ln(820 + 80 105 ) + 3 - ln(-------) 3 That in floating-point is, 0.107752196 It follows that an irrationality measure for c is 1/2 4 ln(41 + 4 105 ) - -------------------------------- 1/2 -2 ln(41 + 4 105 ) + 6 + ln(3) that equals, approximately 10.28055332 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 21 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.07734456168 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 22 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.08582627079 ----------------------------------------------- 1 / | 1 Proposition Number, 6, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 23 1/2 2/3 (1/3) 1/2 3 23 that happens to be equal to, -1/3 23 3 arctan(------------) (2/3) 23 - 46 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 23 ln(1 + 23 ) - 1/6 23 ln(1 + 23 - 23 ), alias, 0.98939253265301134027 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(47 + 4 138 ) - --------------------------------, that equals, 9.1423560386481794930 1/2 -2 ln(47 + 4 138 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (141 n - 235 n + 63) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 529 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 713 c - 2116/3 and in Maple format E(n) = 23*(6*n-5)*(141*n^2-235*n+63)/(3*n-4)/(3*n-2)/n*E(n-1)-529*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 713*c-2116/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 23 (6 n - 5) (141 n - 235 n + 63) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 529 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 713 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 23*(6*n-5)*(141*n^2-235*n+63)/(3*n-4)/(3*n-2)/n*B(n-1)-529*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 713 and 2 23 (6 n - 5) (141 n - 235 n + 63) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 529 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -2116/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 23*(6*n-5)*(141*n^2-235*n+63)/(3*n-4)/(3*n-2)/n*A(n-1)-529*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2116/3 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 7769094648209026246895919678196172914509848199954728822691348330273408674051\ 228289895088386463721668310582269507708169799001475216390480489863472335\ 800001236732930523649970446983864433043071749373582224269293046331609575\ 629322941298890618787257952360030097402663406246170551900249634905189475\ 816490794902938428390500281740275132792113728913038266158006215442671118\ 313153254676322583488744811668704211957503888551804734709380387810425497\ 865214456691335747370384577073376465316413982611541291916485456359281541\ 342801026103221863912981503836828281557716976213490844048011975439927079\ / 113565970636222808523216778708104261764921591 / 7852388603920983951832\ / 966438022556461376887678906845464989630238573910744975963992635631675444\ 513766120965750567859106862573609722903528901367771674105974544646386607\ 887154115408766335939584964940640290140916141966175606124333639780314196\ 746875057991835645963056364065293416408069187897783200987228235806941513\ 117217952734038700522225439737295209818424560870108725759959456120744842\ 372911156479554993584459966212834419388633503678634689756453618976410669\ 482320567023730184098267032708882856624502227023904191315195332971116296\ 001293366824183423043212407971923945488630019962599827861875225644452976\ 096030568898098714182969600 and its differene from c is 0.76868268976861313789522702098451685704584693072191637961705136704149575059\ 321250323126768542774707228072517828885277853849185360688196448809087004\ 395590222309843571301947614938989262230330374283090420584690398823367030\ 491235648960045226323465222543182193225042860648554584215626356487534745\ 961970948697386435188004030274586776358952336382841723679978820975612793\ 326057430521524514503264883117900794841536541349272111425104318301667578\ 444986648134383295268892959664834995779553075809442985182542594344298452\ 440450728773033730502509933154575725707082946609954784591199972771031854\ 688696793538903585312288387091425729381931331709674866016929128176040119\ 457181624865290070340170376375835892808947287848645833655909493708549157\ 798734519497369522533248457239904591570176161230708477025221217742959909\ 860091881076798016592803674773566418965413248261353832236063324657861142\ 139416248134191437421559153518016153761040972493708189827354652745311102\ 267319289542347331571464829437569155067153068444426079130255484469009587\ 297709984856540918428280466558767198368655687435688218433687874215100407\ 009552827338304959681325903998035311342199810204251424051342136933358323\ 550863495039043497611543177247743168301905237332760932171379465632713638\ 956840264189081046345213169954232311152015326609671830216544670605197867\ 846070441114444292959895988860469981081825485731452511903639635935537657\ 204551813582014552009830727204739229105950662862217292414479687847127283\ 429813967612045930817608164173095057771485638626137163709760634701082394\ 137747796579795282917870709228023791086047079145561667041720812821485377\ 501531022743782680111884643624732657174731484950543960303995386869878611\ 036364758181273395669723800487017992185759976057269657806312405497541209\ 071490110498486985965933999732517912561774294266982362355724365819753114\ 789682649810596478243903883867283418270701354857516335076993956491358377\ 521936284067690513574931394362359712557332812596329991193713416657555759\ 849598441763133989030067768868891905938839226376616207072476563947802025\ 742509064725665632570825714239783004805272689163912865986159448734084981\ 323547915272683209473688725493170947965806511682425636674861907701537734\ 882187363251146413877370318147012055831453435448191272213592124114176302\ 896064093605599091896320408717444957747249374948585328351242181413993071\ 150208114878167765981214878708340918368541169415408150483194398698886430\ 244530045615248168699883881271712801770523435531936886296763712605295622\ 491387161666673462876803497416487361419405883691234466286091951928927549\ 805650080497098390855102713107839764607095953452021486842545168420181533\ 886148812595312452499544792412272815228716020452066215832120376150060319\ 786065523633423244966616961476477813390167265195061151606032734968914718\ 989180335636826041310887404342077764664218372457434828760543469552965152\ 574449740596569339274840078109182337494983083141285734297184637689440943\ 493582625068469308371083953486775273936623417499485116796126025070919826\ 372789011054807704075639405278041922310681835905617396331498438337164700\ 850409134347658030986322951548220331915347958104709256919772822657025299\ 549676457924277429493439787675233008789418942854046493572266452565530121\ 100537615313320738241581871433818166099689876418502804397288865152065834\ 427058409072374288000402092512115335307366076487601937761627348559084133\ 812796887045850111231568402832965712272477764203111724223811446133320253\ 105171801948325826973651338565050072484579916590556762034454154888438354\ 198676427119106123579069329900663540416258131760647150734792143214217658\ 408564416480710810518145111133073778402971069108462852389246013293024148\ 734006303409006703686143859806457908920748620586504250920070924553278094\ 956104305751758292218746539116986564144006070203662657502773442532239435\ 931754990745789331201341402090745568108986067432608168950256604967320615\ 753711156974177602764145795458227101638923776713946605660336638481420413\ 906429351806782378363870266578473646150961604743424553525509748481174604\ 161743245712408146698185998019202142012226350061261286143054977313955860\ 544077346001143155034894172435451901979312057854803602308275977588612781\ 229564269765935928412423372500904820739739353464846106983806638311626216\ 040488275033437115174845530486734078581077380107713867866593964900792854\ 162995374751265025943244680204331678408492429792014581096732167907332118\ 102037868141904823026595032208779244683268085566203972296653663273346973\ 646837814578364771057395955003391131603787145546905875332136640787092260\ 699235905157047025957725469060064495644238909570924947972899545183026979\ 904162340524798014169272945214926081618390566429539904584914771423975346\ 258999244611284430440620367414676762991907186114463862315711493762495017\ 341843716343935451288118669107592213699693919953772571666616013940316048\ 204585746909523096461720515419522432591520481821578050358945998720803214\ 561276401542300870503576108176001687364269940433254154111672297432796302\ 238336871130853418782069497503696269198663564003767698216025028842323841\ 813493165543685572947343539961121865689088152358754443729039595929457255\ 148011806513199443628820316445714874803303206066018012317376978785988826\ 251918823034378640037382910609997573525003707751590728018335785315693680\ 021262486883202631822740402805427066629205921593146428417263200995965252\ 705764692308619179082629469246942673113722081328620999276228927532877833\ 132369341652591197240729913116622066018074432316030554338053696514623053\ 347619022814094211280189060594657837671076243634722382943526659907617945\ 016632722941224131190983614788088380903147838040962967135058200478064891\ 329914141318604716687322411870518555223700558265187350828722623732425074\ 081813278688891528719571599433161647252442005687525733999859043595348156\ 631947234856377431353689750863695382587773158461759571573740071359808419\ 118126268889259586707552593488802473555177782044454893563498805225056818\ 352468501253523447301475976890280271211880686653460210605103903466234337\ 271882797029868583932659750931122783777088374699904303782228355644924028\ 685551921506730469037838957717758048510567425207334261258887105620034198\ 671916650878363259227796875233882138352205471783491390448002904704118093\ 219769621663854461371949027792725108161823627191469779351908390470235335\ -789 16191219267371071 10 The smallest empirical delta from, 100, to , 200, is 0.2503759557 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 23 23 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |---------------------| | 1/2 | \-12696 + 1081 138 / Hence | A1(n) | C | c - ----- | <= ------------------------ | B1(n) | / 1/2 1/2 \n |138 (21 + 12 3 )| |---------------------| | 1/2 | \-12696 + 1081 138 / trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 23 1/2 1/2 n B1(n), is of the order , (23/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 23 3 ln(1081 + 92 138 ) + 3 - ln(-------) 3 That in floating-point is, 0.122814772 It follows that an irrationality measure for c is 1/2 4 ln(47 + 4 138 ) - -------------------------------- 1/2 -2 ln(47 + 4 138 ) + 6 + ln(3) that equals, approximately 9.142343007 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 24 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.08897764564 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 25 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.09770448516 ----------------------------------------------- 1 / | 1 Proposition Number, 7, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 26 1/2 2/3 (1/3) 1/2 3 26 that happens to be equal to, -1/3 26 3 arctan(------------) (2/3) 26 - 52 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 26 ln(1 + 26 ) - 1/6 26 ln(1 + 26 - 26 ), alias, 0.99059041617756297673 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(53 + 6 78 ) --------------------, that equals, 5.6071781056925386159 1/2 ln(53 + 6 78 ) - 3 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 26 (6 n - 5) (159 n - 265 n + 71) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 676 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 910 c - 2704/3 and in Maple format E(n) = 26*(6*n-5)*(159*n^2-265*n+71)/(3*n-4)/(3*n-2)/n*E(n-1)-676*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 910*c-2704/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 26 (6 n - 5) (159 n - 265 n + 71) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 676 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 910 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 26*(6*n-5)*(159*n^2-265*n+71)/(3*n-4)/(3*n-2)/n*B(n-1)-676*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 910 and 2 26 (6 n - 5) (159 n - 265 n + 71) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 676 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -2704/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 26*(6*n-5)*(159*n^2-265*n+71)/(3*n-4)/(3*n-2)/n*A(n-1)-676*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2704/3 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 1587597278206596688969092121822056046504774918852075756648672133647595374129\ 418811146924784006932991345902426700788998005000541466759350235165450740\ 642022263147470946815202202337253749935503825482861919557648940501965539\ 791324406023472787512045793331300941543001738686481752006671789906254259\ 473373151257847902886165756723560074565585978663151998366454918616838463\ 776563892324676013617531169570848462485725866399680468712577315328886894\ 034215366529160053429523174225747294474734839170648591829439199288531790\ 345757699313186624115898826669618881863402482722509862811099911284830786\ / 63552839 / 16026778093944535463107560289737389056911187981067995355095\ / 066145650364348967896553776757247531976918811485354083088914335737783949\ 036831667532346338819843622147640614651852268891023019734838730147667319\ 818623136591379371100098604337644285195643859698504408592513068051167872\ 090040404181789260799478828946790635996855739403117212450545829547415672\ 689519538036348291859253820368273589758461355591662588457580055532060628\ 861910380347419200610170042908043259987075733838563546209189492727124201\ 372767265554832664048402547817911866852889744476176547433100155805335618\ 2980480055606135816866600 and its differene from c is 0.10249269823498332650038924425609923680418017537712196041952785338300418211\ 228973943775664201017938280975344973978118760124457212978028047177357476\ 618512018725959847167487193077739155668342937148495532381342421522590875\ 370897681960892129959344693190620692186711078611350039595077954047628851\ 873052963388701053202636974368073504450437309573468327930788224729556477\ 086374075498061647630831342179866965464348532996403544461500843637903199\ 710393243846692047875664819349646199941872019047933822113087774015452436\ 025739761366716572450224913682190507742226835429258568429798747369296250\ 650786179917871504767046269668678404458774669117699615291978355803423548\ 682957334424888405071914755987027011763299444875634431642026284736609474\ 980660408538359921589614926635003280730324050372075419708027320602751227\ 575283967685233847702610454759721324106354446786884905401869879322128458\ 315674248571154202935167602772133700445657470412699009690523492809254153\ 092855884057180314112534650134600279282891159392682987538117895918212657\ 392818020062583862908281407587131333027203561317458394968585912571274376\ 434237974468479203123573639042548335174449173010296967237684513315283689\ 967223960124519496245927633666070421194546218828741614387638201628698321\ 281004202471779602864486100167438038598017290334970390491703022293395150\ 413034203489326536769369024341637286123990725803105132442454351611772262\ 667619354620663102678777917142800550100398131862926381374820873080085897\ 544048194623063722602925834434727138398871989610952366887583652706956218\ 181496352599518260247335564461128289296692264364309545147298656428316030\ 005596005429348285641462940977488863704967572222201849826609601166803316\ 605502522385551972854661838397439347739347800648024365130183343969735366\ 226789833200379536364751766082928260240882610009768287065531590280108085\ 742762692839038217954326431661908261201621290050734392206624307707074672\ 749830104366408110328967893940124302779110336944372361984366032228673228\ 646909149213461874246440063658427671462742612846196523414049532577097471\ 097660681783728482003259703737915988594475554167018996477318765850730697\ 388914079273041431730257401630130803024050845533392678955415107428320736\ 237283380928997504841642216889477994152004066305732241233538544227164493\ 211154675725301376977496595939984724524533622627039371027575832886205503\ 897362417335834213115815501294302316346696358184757360087388445736521706\ 293691774217885397836477336961675281990493840044901032270603541843387708\ 544767040047337930662849929564214794936369650913591073469593272918099662\ 129288369651035598584012874628863639156035459023556622799272722014497797\ 971248463903702490523206166792768202594962067850258036915578571534871279\ 175339850458168282536812195105644328243298874663871920253996974739184831\ 424103932012667895744395047070961366380129737107895578413654187408978366\ 560913277180642102541709706779504064016764598829231340868019412280751843\ 432946585971553243020902869797173109062794601500396996380911765698773291\ 839081249605464959254177006404668099800047918157299031901522044960585074\ 407376599175502164736228616454587392781540323065565620752479563407232557\ 224965222125802859120491148236322788718549178491465947405316862445523044\ 580882492798326080434484965101023151053541561226817307139010297946122734\ 897990939388777121194196677076432629274775336770317510172191614075018950\ 560091589254501430586807898550363466008340647835996272780282601434948666\ 749279978386800518304761965546414808027143571338370645729562427124120927\ 277045883547961949137318253107326179286568553806481475506624872891997621\ 613580167084484162327700015606890218905867273751436200036233108353629759\ 925222329266130610533846209098978339916506931780281264653991145134764334\ 014095804454648618067349506033914929180817312498643077952433837500041006\ 892242054621516962928800965277051996837834994921642245338563138567873323\ 334950576448710310080565718633968196697343584366780918774630363505005645\ 827530602705749670898829979371103850080942964276004659457350560248534586\ 092355528492507801535559468902818852723467531993125392954082856371598201\ 012973060908926530041525532181809942755436550576620968924787043643118598\ 402388675275687537001633831511233149854356007274275728615780968768083935\ 864062550667572559735900474316448711263027225706947292600535354744846743\ 174170354569723621466504318725012162359847249458999762166053915359638060\ 071903952626253108877778226907911283450288850691329541301643989662908668\ 756908099350591702512412629947222116827393329820789779212420054746304377\ 360134523566570730830190234832201483123959849243956161833066791779218786\ 563786494877788959306283840179889264280454965176967537261075849239723626\ 937698969740437960382779592750758799953547589964248439438423309883288050\ 671212977428841419477120349424920290498493852077457273848565574233911829\ 398477604029117935018580220881010516750030081730865673477015224281533852\ 747957443475139371131062658443663828565199361048324462602573968808462539\ 554981620621645790496652364356466314312024365488469505186488516333740425\ 614220165587968516876892431216652772575838089685730504006290817514743016\ 637237359363716850307917184116012902978885715628323785962675138067688919\ 194340113556137942736229490260805311679603438972590439039395608526627963\ 806862548511201779450370858226065029541191373652638375335879161133792408\ 673285567116498241517487241919031266078666455403773485547425664699139223\ 223575294855630257319331617268701251121002332673807948624413233037024885\ 935101529465567185153023904924707752415409218237968975153084891886154134\ 116206016642490457374118009276415808937311824657813125468584398082489494\ 172075828444342274227930490541890566758122551271112720183978598813973079\ 687375519783483967179420639457848099943640269466046695884136291855829559\ 848706428719428553990730953234540085558431272114532657051771612647474419\ 017993436070866575114016795536646541122367815223507062651016876674745350\ 146328739836250788297818816687372604254339507489003140889342276364814977\ 819413235125003767299599251421493796542181801147300158182943871864866368\ 760961297334717522169956707133979962589810395416500221805973836641075117\ 002580967822022395813679535245012146908980138928058385142596509322000517\ 682421178812465371617072774337791047215222640638993766110064093959666 -809 10 The smallest empirical delta from, 100, to , 200, is 0.3624331317 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(3 n) B(n) d(3 n) Lemma: , -----------, and , -----------, are always integers n n 26 26 Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (1378 + 156 78 ) Dividing , E(n) = B(n) c - A(n), by , B(n), we get | A(n) | C | c - ---- | <= ----------------------- | B(n) | / 1/2 1/2 \n |78 (21 + 12 3 )| |--------------------| | 1/2 | \-12168 + 1378 78 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |78 (21 + 12 3 )| |--------------------| | 1/2 | \-12168 + 1378 78 / B(n) d(3 n) But , B1(n) = -----------, hence n 26 1/2 n B1(n), is of the order , (26 exp(3) (21 + 12 3 )) 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 ) + 3 - ln(26) That in floating-point is, 0.217052832 It follows that an irrationality measure for c is 1/2 2 ln(53 + 6 78 ) -------------------- 1/2 ln(53 + 6 78 ) - 3 that equals, approximately 5.607173243 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 27 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.09960726486 ----------------------------------------------- 1 / | 1 Proposition Number, 8, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 28 that happens to be equal to, 1/2 1/3 2/3 (2/3) (1/3) 1/2 3 2 7 -1/3 2 7 3 arctan(------------------) (1/3) (2/3) 2 7 - 28 (2/3) (1/3) (1/3) + 1/3 2 7 ln(1 + 28 ) (2/3) (1/3) (2/3) (1/3) - 1/6 2 7 ln(1 + 28 - 28 ), alias, 0.99124921054387920270 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(57 + 4 203 ) ---------------------------------, that equals, 107.39131966214827687 1/2 2 ln(57 + 4 203 ) - 6 - 3 ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (513 n - 855 n + 229) E(n - 1) E(n) = 28/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 784 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions 3164 c E(0) = c, E(1) = ------ - 3136/3 3 and in Maple format E(n) = 28/3*(6*n-5)*(513*n^2-855*n+229)/(3*n-4)/(3*n-2)/n*E(n-1)-784*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 3164/3*c-3136/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 (6 n - 5) (513 n - 855 n + 229) B(n - 1) B(n) = 28/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 784 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 3164/3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 28/3*(6*n-5)*(513*n^2-855*n+229)/(3*n-4)/(3*n-2)/n*B(n-1)-784*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 3164/3 and 2 (6 n - 5) (513 n - 855 n + 229) A(n - 1) A(n) = 28/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 784 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -3136/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 28/3*(6*n-5)*(513*n^2-855*n+229)/(3*n-4)/(3*n-2)/n*A(n-1)-784*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3136/3 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 2085276818552560120634852810614797692202126760560600597526099399847502415493\ 296507461227574469831862184721309227775354664181065082425802172408532059\ 554480232104671360384224598881031889450085691389722551016412724556086123\ 085191687923692200079517259253706420410320785258932709355844841477437608\ 013653006124341326573720213514635132722247407153557135565946674890109314\ 778735868033795599655194991239613514695481608496277867067148422495509241\ 016599504426747240333482375898278258485550044574394089082516245086034898\ 264522615735367123438504901762423274496862683162151642373979074408393013\ 624491229810689791373446938571094430705278681670823749712237321598003416\ 918962113934528989298714153738183027105419978479314056100512120966276302\ / 4630388587 / 210368572945284789017222033487127571595777607716127166282\ / 029544293678686835291146926295008414700582995283015094015033613390991332\ 446986323757046171592071198396186554239124268850524849422732324421188268\ 553968015178997366981159640185314489597157337896477754050640031716785701\ 911186239670960983199554556815373487727015576106911776839857663359174445\ 307093212512438780493971618306794702593946759181062378315020226675668963\ 839709720198291173163722426650208533500330639422129559466358212154075354\ 351574123503461951238905135011909699383576664517032680102215546982471485\ 761355995345754273676867460236085196045591997991674698055318159339087691\ 198300175994895194675182425418619136088373319203357996693739399308256029\ 76473095781631946800051048200 and its differene from c is 0.23398985312259784181337549414214069017178376950309878401054437798002707039\ 131080002041735866129370858475137071731352131984380335178403414792592355\ 498818260039493577440148114988947708567253352791389586003242016656274717\ 602360334849683976422423413998095264218299782627526149720571344675856462\ 539581687883705830177998808232412913452600042207544306706765908611886260\ 229341041293997320660396936758637979789668235520637683839804820743685551\ 053613262003252212962903062522225784186673682043360913170740919797995204\ 816438332938547563295473764778558136206420829499223745873974150884145679\ 750735479755584804767984628998297136829301011569409419513246655367928641\ 336495525514617240321963066996309849220367229826749744075090790131379950\ 781690303733272927537237576867838334685061681037274808446602383304030528\ 151218809503695078212062003146036453239756854809835799643968178157176960\ 712773270817406345191636818385594995506778207645391077927734581268234257\ 772253738441988918747198364886315309572351973581147114960450293337801417\ 331892024189260354815470242229667388491932332486113796704744830227347288\ 316951593926387768625518102851896690773428649000171051926275802010612298\ 928100674406425775661891106864199832000212713004259615749917979198462422\ 572337982634537266870230085044869542309505724430651347244796760715918025\ 194350931659475551775235708510546135097991967477051465072280270210882551\ 349862559323454469157420789758332096765341020575417598928235206175233169\ 029675447211210830746672215960479335175370125936605304031484026784111449\ 571002853828448613495687325420545065495664118922068650596981622373856226\ 851468530403196656233820988312275080090529386597210117551287523587098215\ 013008564874580859472629826174561642376209139902484265443903131632402185\ 779581912958237858850088163844989025219390679254382390725203217339384080\ 510932037914859299651439289381084727032825888160529765447750248155565222\ 659268007274532783880063224506868874176492714263439369135456708972409934\ 958357035262713949009199961302235042868236640725474940449381568051128253\ 163203245744499433098051457338646557675117673118319305675799997226748205\ 172833569072309751641404062458612005942101075266616151584441262275118690\ 142755048643726178555883921584697604019531363512931318836924277441700272\ 432486865906549478702135464197775670119744199185891279699874094530377190\ 296224294482710070626190743206614543953001492836481148973765535665492774\ 919911951988698308214487145332431352740634425611290004249809696066833167\ 625803282147604396195061035808033944706210638510565549956930036896111222\ 676944606917184398145969092565745838845377167944936041195203395808385876\ 068314027755490760502485429551635649828391902209476543849889144775595249\ 491417823591229939463639696748867028048101441352830106409481776038780622\ 041538657267730980091270573944293967053088886673747683549947682463476534\ 937366162059629742536542229445533284222985497138067382036978131289379973\ 635324179159696642136145823925687910044420837122756547999160514994167028\ 242438572510450019865630248529131646068897371742020157427049766956627422\ 214273311954543623271188746128659653300203970404276634182493998975152465\ 971465473044203342787404178129228972634351655602503527451038773625447803\ 269768174746548260569639902047333731120061177765564064759612927251605676\ 527502176401218546930282814951129115470628494683538735912738920130739292\ 137752395260228476357309188853312427503046506495649554005710929814943329\ 263525400482429639451894678355775664952109917596217813108936980118223498\ 979441616761164770282406674808201139106417382611725817083205547908738403\ 560603228438228501493737057882917787983609542559801740958280313328114749\ 295639555474536412879205178240057929476457442692398168054689817630082080\ 154246760239234374394585204722243994123758061701503999585974475798248960\ 008284934268302193297626891603632709769435101884408735015817083406457034\ 886640073695646702015047044031968172517542369800324132964626393745971452\ 848169836178454301297828920376625160633098280087028517847077569157047084\ 025161091359674755435827303533661585348540709073160930117724634428684106\ 509475460091473870169753448323357016966616367989858437250287608219914325\ 839633809469222242022236652169785979970996050083475228417090801357839203\ 242188157137186027557500217182769605409343983411912246081707188729392909\ 424322097886075724564589549530680868239688738387191811646992935154927904\ 885761265238524063921673429873328749199597946550214261894580014465444122\ 316934828328758832832839275447117879646333966921476847787319368031174240\ 935617023824772370612140778914123391570654735816431884046762469223073543\ 238374031538172368779338796913479285833224951068272620112101706059105686\ 641498606539558382043413142170723111023094612291960028868552328050531174\ 961337051859337051560922502398959007220847141188467880313048812787823002\ 381401841957927691034747743414470258891594038444982411386059459231591831\ 375389951402797130976636658403175642101646164451343725631534014017524858\ 606015035769555000132106387666531119530545696029767099288415090345226265\ 520181185154598178509948155639329644800557524446079917978690880283682190\ 439117156046745297707642986683675604263240282034012330786490045796158158\ 696662878914970615088852928024691438690285903137401770872659559042307463\ 095708738190708015371721323466719093786012376565190618921905455179749640\ 913445125690996106053633273928423306932515736173927097926094754386220094\ 240208165589412179248169742591420155715474774596554645143120056369517771\ 105573537728679024087315470755859615234774077513507750504832503109459876\ 134713248463153589587850319165396034913594205984524931739743824527495594\ 353963368964148666487134567811175215263271268076771755732384744232791754\ 682904892197988563983259584844872594016840682227304517403682382554317960\ 644890654541465527140772396222525889765942395007300599320045723476572809\ 987863069663921517785405730019645106861883850664249783963829683681291030\ 749219983897626152249078129070198027483410388777702220102222423438695553\ 476545456370757386131351972605606223121280772508091317657584829741924947\ 309592820053685118389443908816338780919304603215069824254405083871536499\ 138497445886575488762749382839673836724668121248300186249091722287716740\ -822 35653570054020168632887793837833413845140354894907459866 10 The smallest empirical delta from, 100, to , 200, is 0.1106529967 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 3 n 3 n trunc(---) trunc(---) 4 4 A(n) d(3 n) 9 B(n) d(3 n) 9 Lemma: , -----------------------, and , -----------------------, n n 28 28 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |---------------------| | 1/2 | \-22736 + 1596 203 / Hence | A1(n) | C | c - ----- | <= ------------------------ | B1(n) | / 1/2 1/2 \n |203 (21 + 12 3 )| |---------------------| | 1/2 | \-22736 + 1596 203 / 3 n trunc(---) 4 B(n) d(3 n) 9 But , B1(n) = -----------------------, hence n 28 1/2 (1/4) n B1(n), is of the order , (28/9 exp(3) (21 + 12 3 ) 9 ) 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 1/4 1/2 28 9 ln(1596 + 112 203 ) + 3 - ln(-------) 9 That in floating-point is, 0.009399283 It follows that an irrationality measure for c is 1/2 4 ln(57 + 4 203 ) --------------------------------- 1/2 2 ln(57 + 4 203 ) - 6 - 3 ln(3) that equals, approximately 107.3910939 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 9, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 29 1/2 2/3 (1/3) 1/2 3 29 that happens to be equal to, -1/3 29 3 arctan(------------) (2/3) 29 - 58 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 29 ln(1 + 29 ) - 1/6 29 ln(1 + 29 - 29 ), alias, 0.9915451817393944415 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(59 + 2 870 ) - --------------------------------, that equals, 7.8123095176024788694 1/2 -2 ln(59 + 2 870 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 29 (6 n - 5) (177 n - 295 n + 79) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 841 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1131 c - 3364/3 and in Maple format E(n) = 29*(6*n-5)*(177*n^2-295*n+79)/(3*n-4)/(3*n-2)/n*E(n-1)-841*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1131*c-3364/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 29 (6 n - 5) (177 n - 295 n + 79) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 841 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1131 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 29*(6*n-5)*(177*n^2-295*n+79)/(3*n-4)/(3*n-2)/n*B(n-1)-841*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1131 and 2 29 (6 n - 5) (177 n - 295 n + 79) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 841 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -3364/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 29*(6*n-5)*(177*n^2-295*n+79)/(3*n-4)/(3*n-2)/n*A(n-1)-841*(3*n-1)*(3*n-\ 5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3364/3 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 3662611680716105996666929750803658310777654032101029067853379562476293122677\ 679129650531820633284781154968367110933182476097826801399679917814699982\ 898736262525129515606210408853308578574946202502803405899993645735295404\ 220284189485735237366529585649442494591121063735337765289880575817721639\ 242117874346443540575143786728657505309330814859730345292788247661182850\ 622928669826042930014691297152487233068870974179447788791446520812351559\ 987348600002926410796638316094037577574424609158650281978068647970671128\ 514294318599556810515842443085565159038512286157966306289113561168721733\ / 68601556994097908159859382285277863240223484973572811533184402801 / 36\ / 938424472912641012786880250362765414822448790608977129892495472174623787\ 766913506323201080984521122044429894153072690585302355743969769554258763\ 570845802338880833821717471586243630097041773217844619888585365307600992\ 290557862544994170720785725545894134205162251484638823092717992520461572\ 723583444627501540562429109276831499904687891307833018842487910184231867\ 272081867489282838033160548541266065323312568822196988954252373331198807\ 569592442521117739001684749168122276835479656230441979041951701327742938\ 625771888916121380633647724753646146839910071279523426016552030871887928\ 4927514000355546364905918122254620391775724717266917044493819270400 and its differene from c is 0.23852292922878749637122773163685871390301739793676239058081989565344805927\ 215768759402898937591410199369199130964233800855188655989260094856525169\ 031069743673065458977583281587102340578111980232996644213093050150649305\ 984527948791980364490465691939820948340357669526051128698054506394949876\ 981677626813703696760640106462209748034142562519042714544832603528156840\ 482975402672229139111623890324458844708561875943198644664500122174515053\ 500442049088414457708380817918906769554757939600227400204000813089751867\ 803189965251678324345628918984541937841308537651299836773988580218754391\ 828071354385653740261398410811845638248314534862907247173329665725075571\ 172888179485498727807916259969457695572542576439693681515918451966887686\ 629771665105255333749973550042572845847138831294648884993447691632340669\ 071225351054764483121405712715555587115493214173902112611192968534369438\ 393561939591811105802180389674715191722354447985145742772915161075516824\ 965539912992732757331663515164467618034482017243347083457466494964024837\ 640180244288571636523009751074837246436085104044518432231130022853209974\ 533080176917957785236830536331769987263530182404847681968986930285729879\ 453979225632928021238157908807040364541005080464501046028990478632454403\ 887733561523002081616178661953810027258148150664613126510938214660364842\ 436727778610991154734205702254528369026929310707905477441952281901190949\ 142386570869139543167330184056192971165622388141909277798831878817320887\ 887929838367183534957006136408048320800413703318231755812975958100980076\ 999131344072535958730888637266344257139148762067072753828535825832369097\ 916716563324163008215416354451817247464063364608444848633417563946575781\ 182620424975392401502169473014502120155580603348919112355989501202014589\ 068152466953825355834346879168888142326920253723577643592229872575030052\ 505155731315475571109509789155728441841704878203828692935716726821501754\ 754646288091240756673991457203267065342696386327794877004321978886105519\ 606193735072298549782965220323359400393949933253285651286975203512670790\ 850746679883916232757180246691966250212764382078535471373755499545771875\ 319379947502470355982722738912454899236285884936194921739267156317735178\ 850135842380994628623475622919020018059283588245968696180772125482278868\ 126990518202858440220163474039445640620778520165039535197761147751328239\ 911932725050457466181933030367563437415367101577377082545027298872465073\ 932302625612044164348984528265695408129734328977355724280618791090967853\ 914328221357212225158439248455546019363387095825564433284988554791505682\ 695757053476192906285161758301683593616950553623665485804432960088929420\ 453437292926925686376878016782355794973596857131940758801628145754823812\ 010751917992465756782439782764819488330308826520104168283026121061628613\ 088927082287069415034946617334066233942082136389840486821881455881457542\ 058703649313471229133283962786923271969288484458476431470187906206767319\ 788569125423917381823942591272806163475664766655556906526239255399346336\ 965009638928197759275381558612237004989059345563722766681001762010104025\ 723827358674549574916749653678847742659121870640731842591896435178929575\ 304868798477071291330407271414327470410393914396169422862470983171583265\ 204983356398453509437941475274081116827932517431770148118624294854603814\ 398624125075516071032954736835356459663018557395804113870050117423196617\ 638490569861054256991706688889196847749254095216368853471202331548427444\ 107478078558609897136327976986222911804322869365667715939431347047503467\ 875416075366111679027210962195250385757876193892420611226008090258981613\ 616014949238607937676029689568149067303190592834377353488032200504163437\ 357004464620761035734146728016348458268743315239084113238239992206265479\ 128418620668213286087202384412896720915393550275885474629744451317672582\ 752194292178300326032692166033742775524914048097287138086967445931003939\ 897248144531526898909213145266195520368106025189101967049963954085763705\ 998198486744343164704119066023649623848268540620183709171064262870189256\ 257267694768893351002048749908563964357432540031338670839121663924353240\ 265800505075385481039017394539948646790773751047580808516372428442483274\ 687945760597873956947428442605462067781986598444803635035349112829445746\ 081296683274077096467999374287013722729411553941117707563048150889225575\ 338039076621359583314723272954797052995970035219320144095647771391820886\ 148456733027183073330801416884756902202163709546452649582523144826902184\ 815022462120452222674120282904568677888166278400186313545289115937872353\ 718164775226515142603716167055737970842099909460184806720060942179406759\ 347860001369343810912785237159175203421328403269777796345433361877086579\ 231470270147803303097824597284755837970555622181365700899628175519358473\ 035822010064479826032494226591058528499715611499894830970419604698851742\ 617061371901708154270755138334046676076298136323684792927996046814603808\ 689237699784098997412123057649072653865496399340775621935241003480496495\ 538041549574447925509547962141706933806130240546534195237004716721004234\ 921447425943979192823405928187860571784363975651764883488268830517824443\ 426741352164516536091547291384153790570785544276691409808863041712833850\ 444607177617862218286828053699971693194884243566857174529159124075033391\ 167481471958647603726854246315300453126101759679826296110770319118156751\ 916301556155441152052100331417000071360317809021276977674240289399615777\ 495557881496691552832216723883300949156898839145870598616666960734109795\ 404695268522969669805920160310708188828937468079389447784199784346657504\ 154346945041478391034743176647015431710092358494577572787502014431498333\ 909310609222191610685729486730915203137506548347935254789381792503819675\ 409211141171749868965109573234483397186999312934973290286913021126952750\ 503312820570132624885922338263646928934653534987523109162379461024994029\ 449571224663860405301652639021834104327564640533789036662105913428021999\ 278410108335800882166586283453953624770630246222844621882566647962774037\ 838431030182066819049102883914070313379903430019076676948557221106566312\ 242489218347580001982248060394463924751598762194116512760589584465281406\ 839921879417587365993078639602884551715480815693974386073127826334599719\ -828 87305328704747166865502278718456303616339509439822 10 The smallest empirical delta from, 100, to , 200, is 0.2712745125 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 29 29 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 870 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 29 (-29 + 870 ) (-30 + 870 )/ trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 29 1/2 1/2 n B1(n), is of the order , (29/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 29 3 ln(1711 + 58 870 ) + 3 - ln(-------) 3 That in floating-point is, 0.146793095 It follows that an irrationality measure for c is 1/2 4 ln(59 + 2 870 ) - -------------------------------- 1/2 -2 ln(59 + 2 870 ) + 6 + ln(3) that equals, approximately 7.812309530 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 10, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 30 1/2 2/3 (1/3) 1/2 3 30 that happens to be equal to, -1/3 30 3 arctan(------------) (2/3) 30 - 60 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 30 ln(1 + 30 - 30 ) + 1/3 30 ln(1 + 30 ), alias, 0.99182178558661605519 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(61 + 2 930 ) ----------------------------------, that equals, 61.575171729148795794 1/2 -3 ln(3) + 2 ln(61 + 2 930 ) - 6 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 10 (6 n - 5) (549 n - 915 n + 245) E(n - 1) E(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 900 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1210 c - 1200 and in Maple format E(n) = 10*(6*n-5)*(549*n^2-915*n+245)/(3*n-4)/(3*n-2)/n*E(n-1)-900*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1210*c-1200 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 10 (6 n - 5) (549 n - 915 n + 245) B(n - 1) B(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 900 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1210 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 10*(6*n-5)*(549*n^2-915*n+245)/(3*n-4)/(3*n-2)/n*B(n-1)-900*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1210 and 2 10 (6 n - 5) (549 n - 915 n + 245) A(n - 1) A(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 900 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -1200 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 10*(6*n-5)*(549*n^2-915*n+245)/(3*n-4)/(3*n-2)/n*A(n-1)-900*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1200 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 3307625652978217134933458694192224193073329992617100066945671018569155516678\ 086277710116884546238214728438245075684664137334609103519096516429611918\ 709450333705664410990423522913112243841109641953853514805802190803559367\ 331644510397101525020518376254631065597683020788246352854903352752045347\ 039365503275864710016284424110104257609813547412897855292828357136799662\ 070312690580770331147777924693158698903470854851434069285812171175575607\ 370461301633884454956623022918967244808972447974137699061767956526107624\ 941261012332174321848125291562714759423690386713067039023788015175951576\ 792430916913834539485704998509486907291583114737081886349110307687255116\ 948981139142890911114238913963608157482506138559970341175572809016310132\ / 43378313217075 / 33348991734658376526997485738121618692878965030480032\ / 069618392564444167791209622435011297633196857453942052220499245211651743\ 499230960096136040876372887976728159954804351936490622275593389097961323\ 323611390683801001066063833527058873770906955745723539266812742413557478\ 252320325135012084647031730089899360361804870257677283018520082476788589\ 343510913190143691586432668731360422347109579354975839635789716373333054\ 124942919345817620349065292272876862658498682405443150475640632333330412\ 601888015261902495876083836181850232003459041273826193507001229323476877\ 091843711771136298649638064107343867870009160955120138452111383665802355\ 429087320609081463325332402069837366736511922050603404009762385854399771\ 9774124017329326178046609560518258968 and its differene from c is 0.38519158938446770719445523624198449568067839100760012513625197303906579809\ 478959134867875765320497661817049091859517831428193340598268699017889502\ 039670743515330030866362024706714008270827241902243464966722103338048950\ 661499716615853067938756295097374156770011415159244878660075825949458871\ 700072549298139789325864115902340380078639437418618146764128623136089245\ 493034189619688016432117007537930626228584478789371980325136116990857252\ 196110164332958964935099928516330283676048997153297134496146881018468098\ 172892012554879728465890802936566378061761870251184792931138982355077967\ 344371935778941058417604969356130173403929467109788399330877862112432921\ 087535177223723537379271618324388333400240159124249469042398174338228140\ 517841764500661130089111253382168513556915726426518708486261788287090101\ 930502989798344828277323966619344671060046627556159753063360752243921628\ 609786141156961868563728833478972961382492055615259635736368128492486211\ 389880920561365081415234470655311317341118364881375013679797302177577425\ 195495658251275941269063111211728296541543019811427772637409723260038208\ 469673924406129874622807387338455879480059728930220736200125816218914186\ 014016611010080487027697558165321172981129794969636013699686513469474971\ 601631574865130070302343908101402963025481172500744135768853295505389296\ 563105722918607294111665106837027974795210206076729611048049154423724649\ 835057433638235257492229039707541157885219737560756671429238699400603266\ 546998006171596286754773201190233316362591548903955840887879182879522355\ 049273397515343298040436625336422903503312338798590027675442742699984008\ 044688043783590232630891748702409527781694589591339724051914945410052345\ 894447547739093995253822265440513515975383337488379139391923119400524668\ 602615842354318785423780253584640584368558039124847727459483646951684767\ 900815726492937471828556131316032349496854284068100248478683257731904704\ 613005867321324948215624607779907318315290184732829964663632230986382615\ 968124310640896324909857703355289692150839545537472963297119018139104808\ 588470322771040022335804589558158781964581851543232448849932557955969237\ 439266952625575883232602516782700811486431433428769970243800292755514397\ 653775829807873166882850396060665605740500021003988007797035576676689403\ 589543886240151778152482841488289081786776040520986270427155904311847452\ 405022845457580250432751892775970775549932225900640571852591075272428961\ 311443289700353027984479327837154093064775083753337999902187869680263174\ 958099992615783963282348410418935844792399634156951226498085037366036885\ 910654910668210561860379748666150878203523391396612497053224432065989213\ 111841724331280364795634324911406261581474589683193679937185553601856597\ 922506163688718626518995626001411108718974662538794910725504915476520595\ 385743839201489021191949233315007443758423986190279425839868100837770032\ 944339750469107962084380127846136304459249922927182887017310222015568027\ 954228668304989116859027349088846889752420432981776186235579932608260194\ 939159413671935526793597235120337518514441562673857329401340684273089268\ 846796130644678971832034244834565934861163261349990033689992074725297961\ 994416737983213099068669837937671232003220628257944968746114118516364292\ 437885054040763993083634977996007330789180209445121191753002575555791012\ 594884022536082546856005566346223125061237393161939711050107404657473074\ 154900223730123056714896235634712150856401549522848573038749651581939184\ 984565270592550493602360629994835182105915834827906292209591774705154253\ 307194122725214759725413785085329046734615836625151120798973066583457633\ 339045062974554609023366682442563311748638294962848864970651689901674583\ 374178524947194150432217095779512330887717273202376442363339561850098800\ 461362286366665424349679292349906099487723326794752776845705383443555230\ 989757803673193080527263640574888672759153161354421411175102697403375883\ 764693469291805589925240887475131614797093175858717750785046437654826676\ 431034589432355386229621016134624703645756687781577847987657813473769767\ 908845284749403577510101244285251851552397393097571952000997860101400716\ 618758509588619549541240912863257778775284100724193919086215805824378748\ 327414471892802394558933410354322624581485018151812787463474703735480245\ 892646560406315388841357518401862249426864669697253514758810896372334556\ 743440185128408577878891672423414604612704060496142331255441480636048153\ 482304797224339601661892166447547422256049197130292042854521987730736317\ 869743290524963305686102613394594991048063089707526075097625252550823225\ 006511683077756940106488834018168345043568161791738259171697355737075007\ 801357690810338338791132193728453141601135166756603998480897517652699719\ 316419599053164079105167672140267921529339988764419683304756514698283371\ 731279248170442933911776823232424477777760380529425738980898074696320012\ 622989352771511977418939194990277858842440357954810311493853089131903582\ 335684727503241738146831445105270722523503889729480431528538762843090260\ 574834790418714520082429145780901706603580127828616002925966309883383091\ 102245380505381641250242192383978533132068010038490178903765464051677599\ 601430074445445241524767220556656478536705140243359581184315256496562339\ 371431838543343856402094765756500098920629819089656229468146258288909148\ 458504679394792963588849958728631480414599028648580945176259897617928716\ 367213030241725604490597150359120336678238726859024153480729037256081833\ 657808351419271365590440659482689064429436840104811672838132689197629651\ 745267870221336845826892528873601402436756272396268758986970785534318950\ 515694693983087027221843630071580940039274622394222281205143904991451560\ 117967249133548231077811558692024739632908690924771117763748738834729328\ 064671599902963066444568623278694821186309354465314094028555974857033838\ 969369592577538589816183953835203828076328186658734949766650391964344460\ 263707296295291792152485525953654854499337176801380631971355668860523812\ 005809077902484024622685536393580090925674514744222405961641225242514816\ 969157897006066296340712436783775803437691073836238783095563341208158184\ 779153146296126256004995519352373144330463093741461154696981759658746164\ 630650328485698052178681155998471063463004175570122336933398014010466854\ -834 48304220278229563233850408459444188528431838 10 The smallest empirical delta from, 100, to , 200, is 0.1147465471 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 10 10 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ Hence | A1(n) | C | c - ----- | <= ------------------------------------- | B1(n) | / 1/2 1/2 \n | 930 (21 + 12 3 ) | |- --------------------------------| | 1/2 1/2 | \ 30 (-30 + 930 ) (-31 + 930 )/ trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 10 1/2 1/2 n B1(n), is of the order , (10/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 10 3 ln(1830 + 60 930 ) + 3 - ln(-------) 3 That in floating-point is, 0.016508414 It follows that an irrationality measure for c is 1/2 4 ln(61 + 2 930 ) ---------------------------------- 1/2 -3 ln(3) + 2 ln(61 + 2 930 ) - 6 that equals, approximately 61.57517094 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 11, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 31 1/2 2/3 (1/3) 1/2 3 31 that happens to be equal to, -1/3 31 3 arctan(------------) (2/3) 31 - 62 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 31 ln(1 + 31 - 31 ) + 1/3 31 ln(1 + 31 ), alias, 0.99208086299555444459 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(63 + 8 62 ) --------------------------------, that equals, 51.367037587757656084 1/2 2 ln(63 + 8 62 ) - 6 - 3 ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (567 n - 945 n + 253) E(n - 1) E(n) = 31/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 961 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions 3875 c E(0) = c, E(1) = ------ - 3844/3 3 and in Maple format E(n) = 31/3*(6*n-5)*(567*n^2-945*n+253)/(3*n-4)/(3*n-2)/n*E(n-1)-961*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 3875/3*c-3844/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 (6 n - 5) (567 n - 945 n + 253) B(n - 1) B(n) = 31/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 961 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 3875/3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 31/3*(6*n-5)*(567*n^2-945*n+253)/(3*n-4)/(3*n-2)/n*B(n-1)-961*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 3875/3 and 2 (6 n - 5) (567 n - 945 n + 253) A(n - 1) A(n) = 31/3 ----------------------------------------- (3 n - 4) (3 n - 2) n 961 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ---------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -3844/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 31/3*(6*n-5)*(567*n^2-945*n+253)/(3*n-4)/(3*n-2)/n*A(n-1)-961*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -3844/3 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 1718224988584881404436143982836228709460241921529001906884108519345323842585\ 635721919887441176717327432200933580956259138396147046490466067171265108\ 014148764538838169560433328592726515239432279198159517557581328566386255\ 640910633671132267894953513351222134355993594140268996951389150085780217\ 239864903840173640109085582976932341530948696955215781260371175566472010\ 589177491868764491025841441679160313600939414526328986812038890793413751\ 996329182832345309523356158439607714404221603649079877515691016652070260\ 045482289877710376092681546089303133508838333795978678033107895835078524\ 348257242972666650151145998678400240851560632997549387421266175558855812\ 085745908692893011334864504620431058661712054358665978138812754160261625\ / 878142825987 / 1731940462390091322360222032128383785564605445768342470\ / 575638300592883433876885950758809344574011449549209105037120779621957212\ 677629203282500115878669039429102473971051258998308242145268985917225183\ 630725624695459991266430080584872144891849235477658302096918342541695521\ 267794075555263094116024697298859567787507734028103815551053308972298131\ 552978417809373211066701885404876643208091437688169419482489347607463046\ 954240969698561092484517408659250726279294113118789278464518123130824876\ 374794426035661714943973314765352216508169280002217582033289125721641457\ 690219945146701821746469005418001684673422266813531035215608627560401371\ 735341297343151679619781687460176321413047507508996292839843281972701714\ 296073090412178448036252560364800 and its differene from c is 0.95661527478836409214266610174866502909663783896193488740914478859106316664\ 251792128249541101392455697356190446011126039365150397846534169385927114\ 360894437691287553375085401305144455604983987315061071693408510526340441\ 103242905089804394891298440700418258593892951963450102641370954289491435\ 124967374342574766150761481109782921065885097430147051302497040742429506\ 992852162196198457915276828138815212953912462007417172082890628664021997\ 094457083438381327003429343879094521709254685876053282144384707998054053\ 884450313103201002770213101105110219277307196978615342839348848434231623\ 452147779509380882459072468905096363645224415625459571268775793016310475\ 682029199106342232850537101856121193395965404935669505027992409167494321\ 674153114789631867287621241370183568320743528515408063913288469654331321\ 734383261885996130789873635104216010145913542210381717694173934830287687\ 542196609429695220237316355701298338568392740364050406961150862215594415\ 736378638916757747450473585734670411742681610765329322226568860782747046\ 018188024264836999598921885928615035805670513335390849382507494412897415\ 761998877482890810866626283236666400601092057229053388151259374964790107\ 432503195355398593074132223459127945344941403735252794112357902839938737\ 566188729674809345053230359515816723790129456843546695974293058008962615\ 391757795096865663479052443494757790819911468380268284028254007557134916\ 665198888953983537613044477454670744600346556964381916812835399056325040\ 187757482869939973251156005553806020993043323622993595732229658895802414\ 350595437722410914872135635597080119594940754359798758850676861369176829\ 261148678622733483426313638191418640744603720161885069847117182560080053\ 733151122378976788494924659018183636477580892405106826851412909832960738\ 030512018113121061002771047670948053135063509315451803875896892323411230\ 674548963992000869007155981099513225524696681424819911798536189802234165\ 608891819305438793596268271272290539429796035769190924324005986689617213\ 017064994083402640033884650185861515880668046139757837170753184364816653\ 078812512306177149043049351844363957029187184151598670284363028067303118\ 210318740996723475037404243320007766465726002365008888268844362835818024\ 415120028550762412014299575438369647799014572537712726991458057393662339\ 442667217860740779608550544186090198334714458646881183157835983728600215\ 727713747798714525971445382024087295022836967599116414127492446739538159\ 013040426044569900549648405490126828736742183561581041292568856290981662\ 929696678368999743684504574155184259138631069338332571574331174444676345\ 960053384386484820833256882484736627947296640913327636948848442224750679\ 503099131079795218920985829670706045166662882417270458386464294742011550\ 753130743318797226460813621840965887412993809181090876010271743018598016\ 121842436439444710657667976288861578857899115184812799463599551859026065\ 772244130898854193068923178250566361631629589845281321100547483988920676\ 033566508561748965740079533153472793397391759356688064530845919805696216\ 730401043855440457025295409568914950031747955022226935103383096443474179\ 711491369956148074132889976678805370662106079104135972717491289754121121\ 150486036736881257990253117824419475889655851539073642846314502267585566\ 156019102218457174908767967581769440028010749985919849578637774905614910\ 817522341693791736993573687547638369497688457290363909893356963808924508\ 510071530314718611466655172324979727901245312308528819266400108764932985\ 617200221221603330009798477703369370724344518481722961823166425713555806\ 987137135070890637896171784341643473555663322517106670086895608824709017\ 723797765480511113980602717018730131004330441351757709794436255168693702\ 388261092590244148942837360858534190690635372103145140045618471203773338\ 534979314737950455752462553699959600498847585469839486131747625197996321\ 869933907788389476817430236836231061155782823166178709617514138408741991\ 634813802803661270395858243418531129073126396480226123035364109701217379\ 192204532243926477654040767153988961878897842333042167842489716507968896\ 017459082138433124579242427770503369239729959099874722272957727930627693\ 890042390640121748958027357914337024666504794083710019635752648594490004\ 968604350021355305635422594709957824118596171836141658876206693237698507\ 439997862145053032630185771094029923483010068548713356119945739122696853\ 100163306290845583941182690972040297949457719183628121143483641827450478\ 549951170242260138442902600373364029391073770664708414169696951400022981\ 161308368226011439348811038471505768190352784968951519048717971367100217\ 492153327357450154647297281609163644220603441310845543673752508258890332\ 785170465484166370741624712596299864677393171430478028644673179241958054\ 272856705542282307826167703961128198138294210576767801688678544091506708\ 042017388551810878478225863524887941747129709707628864106055620564588990\ 778003026596632879851963919120666757201482380697455839065275561958502792\ 642205393463935204146569884526185431280835432212172167774756599049810707\ 288373846491346612115339043792930051638468775948080512366065919396135385\ 391349584574020517430948207190963378660358817563064631989411667397375501\ 508685395202563531660848494388833222313641979504087942891392628225546175\ 838976403745483752580455854197010740009060593899555519404342142039548433\ 637448039705866887650496872527133831360657291909547567887896578579344331\ 919273797433618864484227113122614977345062329363706290435297679020348490\ 241853161983300027790254209438497543639848720651133697710811729332001890\ 031987164132875083546258380448617133642461712895900082517868453135864098\ 691370485234957052599284856535319111811721935828109834448854238741153470\ 606960063549649440942596327582025596078772663198008821720345643601657582\ 506495691983837352476799601872371863255988517960735742132229338733628196\ 496921357281532724011428982726841191690622079863419564858693320408771544\ 689406587384810805585219331658402216264256990287729389703311018288753041\ 140470777214451583385690496620399395025571570144148602651798622011485322\ 003255116303715456010667962655460808304067780599927389932494357382360118\ 217832258506614377790415854138248778866148137841219051424004060236324713\ 081969583263669860295277113863447530321559426800716164955062807975221836\ -840 21180291024488874318689242757917814135 10 The smallest empirical delta from, 100, to , 200, is 0.1170110786 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 3 n 3 n trunc(---) trunc(---) 4 4 A(n) d(3 n) 9 B(n) d(3 n) 9 Lemma: , -----------------------, and , -----------------------, n n 31 31 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |--------------------| | 1/2 | \-15376 + 1953 62 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |62 (21 + 12 3 )| |--------------------| | 1/2 | \-15376 + 1953 62 / 3 n trunc(---) 4 B(n) d(3 n) 9 But , B1(n) = -----------------------, hence n 31 1/2 (1/4) n B1(n), is of the order , (31/9 exp(3) (21 + 12 3 ) 9 ) 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 1/4 1/2 31 9 ln(1953 + 248 62 ) + 3 - ln(-------) 9 That in floating-point is, 0.019854143 It follows that an irrationality measure for c is 1/2 4 ln(63 + 8 62 ) -------------------------------- 1/2 2 ln(63 + 8 62 ) - 6 - 3 ln(3) that equals, approximately 51.36732132 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 12, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 32 1/3 1/2 (2/3) 1/2 2 3 that happens to be equal to, -2/3 2 3 arctan(-----------) (1/3) -8 + 2 (2/3) (2/3) (2/3) (2/3) (1/3) + 2/3 2 ln(1 + 2 2 ) - 1/3 2 ln(1 - 2 2 + 8 2 ), alias, 0.9923240287154060486 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(65 + 8 66 ) - -------------------------------, that equals, 7.3852059883942064344 1/2 -2 ln(65 + 8 66 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 32 (6 n - 5) (195 n - 325 n + 87) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1024 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1376 c - 4096/3 and in Maple format E(n) = 32*(6*n-5)*(195*n^2-325*n+87)/(3*n-4)/(3*n-2)/n*E(n-1)-1024*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1376*c-4096/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 32 (6 n - 5) (195 n - 325 n + 87) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1024 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1376 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 32*(6*n-5)*(195*n^2-325*n+87)/(3*n-4)/(3*n-2)/n*B(n-1)-1024*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1376 and 2 32 (6 n - 5) (195 n - 325 n + 87) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1024 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -4096/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 32*(6*n-5)*(195*n^2-325*n+87)/(3*n-4)/(3*n-2)/n*A(n-1)-1024*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -4096/3 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 1264507741833737456336315422505319693730816626889554996993156598796417728680\ 002885324987122421117571734904657163706612921808496054760714228198749978\ 334945164299615496554190336890998691631146404961558233034192282204706081\ 591572062788598828949760084100840586418372615600868207779091190830127904\ 000114025248975399157082344949229751267964544102229356741204217745090011\ 099056236358589628794136258115122426657771693629663590263063376341461149\ 431265972965682401535969272481749698270095636709885196743432739499888113\ 422182129102600097797754201730730398095444482661902927164908480284832783\ 760424003148141902233377899300810841416226467368825618019994595337370392\ / 63 / 12742891487477952026525965364865842549873710705704896784199597820\ / 749654685608728676691757263121468737971437925263510414688026767183547249\ 964309722072988004176484369045829493869130121547050873977301633937175378\ 388184857244972825779463431477905681847310356480947042839973271580925755\ 453408961545143911557959598700277765816729308889752825283460472896825206\ 005470008389215239504840524469478674701640477217525654648854636583403010\ 435307124743406865257371902102134397663259759690772406332508040558991766\ 544574432524911694207018014460226373240251859322229291391675503412958135\ 007902579895423251946460631226786146697352472563854370707745263027364824\ 2011894336175 and its differene from c is 0.35564961880913770076454811848859568902694896251013595822205480642788498843\ 362907343163057696671604217306783444246249280806453419619042997542804411\ 250302211769340263513141150104407298332035123091315906067258846066540667\ 088816598012318803419418249542285102894864805094912581520233359901160617\ 507765167898499261803894777708823213963165833492007110618551916942142372\ 272304091277385179171022487986275361641979829107688708930626522396724138\ 702568413743879440317129799377954423600239818006349870176343087276105163\ 828785085748756445976086811664133520118851336619757296220663933843752628\ 708345349570211096642528642612483875728316298640815747316974922720163595\ 643851426869652201600957821828836489950826828858455776128773178941375266\ 602468437959882854484463387846476570896798222595849066009998498724268340\ 468328499248560449831108043290954704856238451450429563485903703461997624\ 020597315805593480991821259510175223236036825037802434891211314256735820\ 701887444333104047523301600222184278398074314354672217594819132142676646\ 328617600812275296101209793512563876145077857521944332038840168537290217\ 947928286797411270282389077261915046881620379814636445328718624640694096\ 075833436187545065087795023630108102581698290086896947731492036574044450\ 439824218617070494995209576399233357748093201996676044915183210753571870\ 746264203626414074677440099737181187774800769911843540195304142560227991\ 791004140902613716969944756615069625374044939323348789215776182696919339\ 982792848695735489507218863905369272632711659054467988826001379391444047\ 905899793890797493757413203774139864382094065703598762483736525394724155\ 306795876197565585897444658241702472681439482751262332082695705339668752\ 320721392108708114060323745393853916275992359850264857941805555422253479\ 140903902293592323721063473478506788588579123634441007105191714525923408\ 073013645963197385454321529333538942829415804614557140873797779048077780\ 052307357216010861281253472043577155106463393379196448819837165384602962\ 150748452069727460777354298034285299652446017880857862383731459414768441\ 938219407734766865943862465985942668498366647836934228024051422767984278\ 449675301908666422050197274335916068806887535226310352919061292091970544\ 886747360790722968827065392251306843797963074951046821075483726629938925\ 935034218789986714995541358910327999301875481804532047712367264638313237\ 190998499828457575002499710915720656822207826441225350444591755257848598\ 538901232443391553605037772663361689552792113507048344166809396863415826\ 217617559979890177198186532147579502764088389371241996140933589549638437\ 273191285447982402341104492926823244359250758169550530614761860677597064\ 611756807718463353349383872041642363789075834414792930542363218225326784\ 848704011020244083119287751276739708038275697148743665091990490631723046\ 031551693274551865887270617287722922660382690746209552290879724895203110\ 083698165762116825621658378584397693854951328512001634907955438000149208\ 531335493528841871562462166698299410713421544807781955077647272152273086\ 913067383209285145175920986226937951764303149418958526697380877912441616\ 263501242478043933276632194468428074804612214307952895377668707761369612\ 368013470032103480519236677629796986704005519690168414741410937541366989\ 746330555001483127079073836947464642342645555349584128466944949005588335\ 797838920854184092259076923575471150023181193770133588450574065431116562\ 635290947549548716294261278756181464776134388514536343662142035743451163\ 682722412096930867305980234229426330159703677394001051493280198742208340\ 472511859071986927860692708415061099520015424269759156389760438912731272\ 860353215860590318639720397825755467947157827831654906877689711600088395\ 436930123144071338913571196201680214597521709552792091747687767691656658\ 719022040441187583162090575440478891130261431272843235750025400284855904\ 791850146633404376800639571676834997454545633090836816067419566011278893\ 768632735607435641529732872382448149135632707149799614972763574511228173\ 951635212340116885078996693994133464469229906455324124113031158988224817\ 427664973677508807869442040558476716500592460470157085561723855956359754\ 089939526449045821298834404449236308263211570759730249538475141053863432\ 455923600789682736921392520811242637199052502354154658683922698428511285\ 194842699158226825331734158270994490904235359519573170481443853817763408\ 022577570403411572449003647941390669076617517764707057379440010996330180\ 035810890995749795022647759751506096286757137950234284238769504156740493\ 071282491618282834622792505523154150637272428138886274776556808507028333\ 269135826915365790974691007645920552334632270541150542891472476523305998\ 179014205718198772441281748647197876951698062210863010144548861905265576\ 228835582212920394563052100724961059156418600535777225650251151533765309\ 193121921101823900095801537287599658853244706601167498011276996298998141\ 770791605054032447665254873795677314776884708920192083315988383548246758\ 329716483508081735554427382920636308856046211407135355579389878794381861\ 355827366335599995292050561017255806994270250291328552810389264295977312\ 208513687490299421731315425375475858755234663599748039060030693591066709\ 511429307205405114428403541874921877772951740389769919357257703406137016\ 878345490533684580171092694656654770980061881185793568407225584380509252\ 453827119011875820451578170475483712955656241371327461860659705179136881\ 568491620605471022951008843342785190223462431274654384066015276879166047\ 872219695740113889564242883247818715640292926208455546606472679256190000\ 497283887113347805795893414051656500531622192091850200189073935462868552\ 640655332713510777609716839010708444188059368235855819473219716214544361\ 243593319881992505539920542569040901499302416168300165107814573877032897\ 803234252936384643842331904296977503361819912855120527618719974203581000\ 831284566796419080369237958879899909252601139166619068637258423567205737\ 636855425646577911193364654991932322189845509338583440818923685746653468\ 434887807848906493062691684100923358163365634548553901625117493022474374\ 223851595771485795843198282525950843295107535491845632697143739569236270\ 536475880833015612527215892497298843318272773390554206780163629785577392\ 304001153359575204352931864967632393112163568543064273440737215234502258\ -845 816739907339808868473751542389213 10 The smallest empirical delta from, 100, to , 200, is 0.2843506927 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 32 32 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |--------------------| | 1/2 | \-16896 + 2080 66 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |66 (21 + 12 3 )| |--------------------| | 1/2 | \-16896 + 2080 66 / trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 32 1/2 1/2 n B1(n), is of the order , (32/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 32 3 ln(2080 + 256 66 ) + 3 - ln(-------) 3 That in floating-point is, 0.156612116 It follows that an irrationality measure for c is 1/2 4 ln(65 + 8 66 ) - ------------------------------- 1/2 -2 ln(65 + 8 66 ) + 6 + ln(3) that equals, approximately 7.385202024 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 13, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 33 1/2 2/3 (1/3) 1/2 3 33 that happens to be equal to, -1/3 33 3 arctan(------------) (2/3) 33 - 66 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 33 ln(1 + 33 ) - 1/6 33 ln(1 + 33 - 33 ), alias, 0.9925527050324660332 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(67 + 2 1122 ) -----------------------------------, that equals, 39.203337471271641322 1/2 -3 ln(3) + 2 ln(67 + 2 1122 ) - 6 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 11 (6 n - 5) (603 n - 1005 n + 269) E(n - 1) E(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1089 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1463 c - 1452 and in Maple format E(n) = 11*(6*n-5)*(603*n^2-1005*n+269)/(3*n-4)/(3*n-2)/n*E(n-1)-1089*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1463*c-1452 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 11 (6 n - 5) (603 n - 1005 n + 269) B(n - 1) B(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1089 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1463 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 11*(6*n-5)*(603*n^2-1005*n+269)/(3*n-4)/(3*n-2)/n*B(n-1)-1089*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1463 and 2 11 (6 n - 5) (603 n - 1005 n + 269) A(n - 1) A(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1089 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -1452 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 11*(6*n-5)*(603*n^2-1005*n+269)/(3*n-4)/(3*n-2)/n*A(n-1)-1089*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -1452 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 9459218620069452034230242849079350546462541259088161708294389060583736633302\ 067383184300089662168592450590415761482852462716359709102896894595321013\ 172274351225691072019375683349276070746254375721403730508273130313801742\ 203531580915777868013240938807301203219494223359446594893400607366603720\ 114743905401166604419947316962988977707514314532570276442886115235062617\ 282849387290356355607550030761333484069300132488378105051269408857934137\ 616952337604051699535833713526549113281163477888366862192635692949908331\ 826452000264435086649783198743684133572715023088419368050438525593428738\ 483362758631540464810727908873539254201123513910582996740378221098917107\ 607106944306677955786811038377625567477877472728079852956658022301247024\ / 88752461457594274893825343 / 95301927767755612318406615637553862151430\ / 903910124110797455939235760888388566356239673727887370356246889936895848\ 033393951092483242523698070615762916888997502228626793837235008251143523\ 709076605465639753487825791132087501573497313038581786449351232881603826\ 059649284237610324879419992679100657859645923860206482066105171014618344\ 710854291885118227541813793676813689007327539493314647814828001121067318\ 837060210154666067417887471151417009739119500622941375747513597975504758\ 074770549452382770908087940045604532075931825207505380585712518981991209\ 202802440108187150315475719361420683509009193027881580856874979593752721\ 937168708961174205304714999615633198478513599990753750611244040430133094\ 8674389675886684619088785342946495726345544189651856906028800 and its differene from c is 0.19315505511994802755054561432573031905721074000292419968645346886657010070\ 608575688090324146350949982047580573198125668095660570068473707468771556\ 696186478916718381808338813899287925629566587542957410132805358098430880\ 671028103957629469582669183957366676437779853694040499959722554050686365\ 935684893270270623931004697435240822050123591956510675733382595084225273\ 565506280265823509667223010098496300250744184984702457190004510639580344\ 548013983559484770973830472951588202614507113322800270314587014286907024\ 398144686983767493617727476271891456277693789988801157518578556180291739\ 003857333412665307978220762525837548206513070329210911212868492140728597\ 618845681897900631484980131721051376478626148827620212954487704656818937\ 420368162579364226324324178574217004916957967536784000680810861793103255\ 227255885444588280399478035273514896958879664202705591790760952067741558\ 169814237718321419961766006478533543872412864633589361035728485704836432\ 097018648143971586680178062661928379978913098226527649532388687755126620\ 937783422643364287877329058765658601911964147163062844587224594894489263\ 989416507972787501994998737631918839378370739004798309391499866966264250\ 778044942460234894399726077929835025236559271811586180931697397401728746\ 124752533953431689042793549207289966428490524263586550000054089786846106\ 508789292329607624972451890456291548518322262794534265250725450602828398\ 653285098309176203841152962723414241581682643505871081323653215164289481\ 500170345133855365117421411019267930994422672387850113997263073425881719\ 316635991305162019494283219259791216563298175095639891385944571275889996\ 887428377974426202022141248587381929481750121796819569355829966012337374\ 534774262371942957468056977927650056583648305241629385771049996887063266\ 591984225968797914000657881228002116110283747686067027134873526794815660\ 686020391818567921301248284887342761101600437761808161576833353429780581\ 333883973912394771772167873598494985272043808271759699237314586076452651\ 693880205285712747684513988703218705918416231063705501033742556172358812\ 983606336324026417280167259852171752242190859632848338593316039868905505\ 815416808817537820051420083812613050822754678078652833948880209745273406\ 878345266611517587914862997957182636741009712640807427442334704924167509\ 572342115179414611796016521887043699614914631725737113013818300525037467\ 763270782420027148121436260461844756672374066099521671531655317194144925\ 267923415373757866515613730169184892522973315498835964445720138513507768\ 574095468947344789746681705572732081432076583681501757412688628329722865\ 578082060027209465657226262951331142763214227656756002673083233786820455\ 409273393892935633994056080185510261410648696471234588741223712092270295\ 953137079983322731020114622523757243364783008824821004059134290705164893\ 537422689361046697466048980015619370372234710573142163465506677584498341\ 659347797431799058179274655654685377737312134852073591084341326872146163\ 111212059856267756304174904726970685987261646687232555771995514808553212\ 676245350675233669385123821587094966230260997024467380697043166860806584\ 148819184400403649965563658007487773744461234738033205678229898319981375\ 955582421534063553493551291179820975746645840868270699653118094281289470\ 350491336576037447692078652879167201324121501784887174205574321598160147\ 769051617428952393765440607350258567389108248796159990515554021819339560\ 928897299725661624782392903739627042211519545431328408394576308163798840\ 254771222259846523690606055027524259466463309629515541700642301667075747\ 104373728419207235874341419711902661261952676460965828547653025886464649\ 168150369321537573007008954272935913260831748349128265199494222002847167\ 131596763367668169634066021509678276963286216647732314165811106797513150\ 409588909753770447437372974976390812502897601327198139967849743711678065\ 680053165414274641096096159884456146106480117997908682564726480465862683\ 137327360849510402928403591685169409980662320624953449082654217650820926\ 471900638288660788001547053092015718842913804570732866270021911750734665\ 652228690070583211618417896267346755355390585191355064712513024586847824\ 639475799676157195841211665844135393523659835452346029679619930179045525\ 258433195322716306545576559932102254711923789161170803958513144852764860\ 991864621649388669314487438631929627115319820225370847524899797373134171\ 809854219868833817365173479686686652524506319055698123917619955570897275\ 946424496190605800317338155361450879850259548289205151877355389667719176\ 684648752744660854681354078781458860155067655111474303146802715424544722\ 029475642759966964683549691954809211582051534971350896280080622414915304\ 396177852572097366018058752631535313740338420260535535632715908142696114\ 186822627905900349434756036314733248353942020305002234234227480640463557\ 844603382947009425830857555338154262182619305252125836314522353338962897\ 439989921152447583683819857394528556313753393512739062803045661649703003\ 005213274839205037404304861557216240523089896644034769885219436091033084\ 281755203857694884419801037358651835571075131506191890254062867393092006\ 948256194970348342933311770223817683866790742099944833494954254505297978\ 450941888132532985534102442983436099554356499478526656966393347336991203\ 431687742187314207495566047757852057017909461109963906698101119339633674\ 612700510210401248719892864382378766449274528137298765199157255828950802\ 427772782389303070248608945197329334220642134735831073098378237009909763\ 192371200165865516143155309163181016710543689712855497420926848193628900\ 027145100339190138897363454853115496711839455104229668635124998764092741\ 283014268994831101801233160358464439105527914951400589612607176748528226\ 929351778477193293033630518346614066578412376422391259129340619452943074\ 837370409205834458557535626759493361261224328924443344348671575219200265\ 547640924225078805675306995798700641963102334580114556116205924568752769\ 566240243913680657180365266492888429446943085691768793275358323525440487\ 921536197467413676012118491061996571592937151091641070401011367744410257\ 766976415183954872574345066509471345836541430124177527289201240266202150\ 241244458659359673137570551822971115799219323210579283385039316972500345\ 301899554556982652599262499245152476853032027785135333809196440889704835\ -850 5784198396933635006639441662 10 The smallest empirical delta from, 100, to , 200, is 0.1244039007 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 11 11 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1122 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 33 (-33 + 1122 ) (-34 + 1122 )/ trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 11 1/2 1/2 n B1(n), is of the order , (11/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 11 3 ln(2211 + 66 1122 ) + 3 - ln(-------) 3 That in floating-point is, 0.026175724 It follows that an irrationality measure for c is 1/2 4 ln(67 + 2 1122 ) ----------------------------------- 1/2 -3 ln(3) + 2 ln(67 + 2 1122 ) - 6 that equals, approximately 39.20333680 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 14, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 34 1/2 2/3 (1/3) 1/2 3 34 that happens to be equal to, -1/3 34 3 arctan(------------) (2/3) 34 - 68 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 34 ln(1 + 34 - 34 ) + 1/3 34 ln(1 + 34 ), alias, 0.99276814961685250853 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(69 + 2 1190 ) - -----------------------------------, that equals, 35.284681076908688100 1/2 -2 ln(69 + 2 1190 ) + 6 + 3 ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (621 n - 1035 n + 277) E(n - 1) E(n) = 34/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1156 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions 4658 c E(0) = c, E(1) = ------ - 4624/3 3 and in Maple format E(n) = 34/3*(6*n-5)*(621*n^2-1035*n+277)/(3*n-4)/(3*n-2)/n*E(n-1)-1156*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 4658/3*c-4624/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 (6 n - 5) (621 n - 1035 n + 277) B(n - 1) B(n) = 34/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1156 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 4658/3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 34/3*(6*n-5)*(621*n^2-1035*n+277)/(3*n-4)/(3*n-2)/n*B(n-1)-1156*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 4658/3 and 2 (6 n - 5) (621 n - 1035 n + 277) A(n - 1) A(n) = 34/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1156 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -4624/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 34/3*(6*n-5)*(621*n^2-1035*n+277)/(3*n-4)/(3*n-2)/n*A(n-1)-1156*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -4624/3 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 1112802579975972813223491426069271122291071544632657526070885727390342006803\ 336779817438326101651788190902966613093465013275629829980605619467423420\ 421307803800840812032732428391068139807319878301277899544553707191815504\ 212546296835426513080210537330332674150275147285235475654325446154354815\ 853739130493046445383700404556647243470146731471780042642079645219528559\ 873893348882877231611125814184933794356758222423654678170147346895564491\ 970611695749963267387497871962939911719170055591564320258919432713830177\ 209957915723521840589271335098496710673818058253313238915920788121134951\ 552849315911181420867461368073648877259764770570905345478086896966226078\ 241312586943036959628645839425328484146248319754212363340800623302764449\ / 2883966415001431282817082607 / 112090882489073225958977258076298912668\ / 012786726347776851374163591473906347921472047142756572624856968804465396\ 198420699231994774199670651662097259974765142731628940852747296009147334\ 247461997955632993529304588415042569858744529775663455195458096516344336\ 046607819327686725341101065641113661322276846196900114453139200236076252\ 329012145793659946890036035082638140807899646029312798548088838609683181\ 114810394059912933163931689807195497387744880054676208142343235951303919\ 467518944183088194697187482212190037992553223643146107267761402848827691\ 442480761569439628430059596637598016705257402472330706741059813469042254\ 464722939167278485174787281760456168778379854618933064851208871564526436\ 08861614623199736350142603123185853356840045871232451272415005800 and its differene from c is 0.14986525195451163414868205165098405590591897339974275977362364117801343451\ 416691411918206537304282693689847471210809828391506931594655979766669450\ 719423124022573580871832466340853450400353335281632262405052620885893252\ 522441204160736644274459122075319869668244405538944895400361686364464886\ 354627590858349930532494738941200476250036775816148582099646377455204979\ 970187090766377826483871510435150542530408565805875476592252592703382434\ 346234927840370108661968449429743730157071408419892041086169308868488038\ 378000665336613308424084818213676633936766975161928101266893555048937341\ 507189307476053363857994736078782438590369905125347408095896876663362748\ 863871909273807018302156900096936910902619153383074238671942485391310382\ 704979121247770475404875842990373919975764754762916548701889424031725193\ 992199519272877694815591559239971123533506079354376894090458604713480352\ 164949612303000353723970255799292408900203789268465901188062800057818167\ 067085045443387105869368055328120900668157126130834272165880831210072754\ 340615240526309742327990149268121367028737484201401933478826479102350285\ 896876025655505696899240884646460559244633809745077114786756164515680851\ 863409581280133773931499667493566456357159805734379270551247298530186939\ 805879570584467529235177790179746626990505143614910136229256188612291621\ 871415961052107366167694674868871779707412536533054978192960923865171181\ 384558641948832497442210294572887336429988179422385228549016637080432847\ 170089684740215475241988392933430313127656157983456959107381506320615584\ 665528093072054521550065275820984879249972155204275684408374932840590709\ 638644079229075449895254175348522678010582390537687944962777670281557959\ 183192439043173153213542197725045641107831673487130559629430701900459871\ 850559146337182184888420453481471899826875792920833244427359900383333979\ 383891449997806820586497316176103610079756556920166655809060457586074937\ 603124195981782957687470432605463916434722647732402992164686961953108127\ 349891351954425689189247120668006704996520560265896228455515447765155425\ 400679294432205529269088968014325101396815258249684180167875154522365733\ 769196357455395311800866485967177366994256200868010407384974802244576826\ 011561096723667649403374619366950451166511181278061986548668121659148486\ 959093868691171104292197476829694850506913924456707570708250946447762353\ 321608585284176384313322522654339699861561053777104780595501618944446033\ 152693571592825207321541256050292434692208586752704094450229713644399371\ 489432398561734426235692327198284344699937172401911862160147110793765928\ 517348025570335001395179520157035916117493066636504407152870727164256551\ 687409786340164659496593524643878092863659993412514203046264958093459985\ 427671452034019005669806799967570342528114328065560183699281750694459493\ 852069206928776539564064052844154343591096376409029605247636518492619397\ 291985798123742860675872060371637612541565379276725363909286654468088023\ 573800544092766483074406047140026550020812770957564642624741319237751064\ 076136381765305786003349542342658672163926474300034193174957242166368605\ 173983638748686634848604922931229495114420355492714660570837899470988916\ 512066165351579229186649866784033498907244047792797563470378241731205509\ 998136876941401693049589733130335042143532968622605941235704813510082985\ 565238880889907140878653125710248176460298280997151609705136521116576652\ 812742694892602622964091963247076665390011136091975818166936056349351189\ 473770297756342927116983019839774506250724365668192869143516954720059586\ 894489068393057098080186809934880314904211511338206540652452628611349567\ 568968858662049472109513554165131462345161895475849722289188521998953210\ 295933087817299666006891826942942071284194937036980066644878046396332545\ 883021005494950728176013652463992107140543375695444061325298064014587064\ 179085944222651443194437495947603745797709538005510977428524505172272833\ 625856706409652380007228771498852634258354405206143682298828498307578350\ 903945887056414158081001399085575673912414706928369380075696821220194443\ 376360689986514478032352140795301531065773523175690028590598107479630262\ 961817410860395329767946242043142438425751721389087371292036557454405716\ 883485863282295325287290580120583408913703810048996157830977547668223246\ 244179002040509629011387063279977319970123836948679899545104013684934572\ 831989408178421676515793140145844293301694267739595303784130978621728933\ 946080530418029567018310996695656845756210556366611343969920817550857958\ 230933519109177571376930469806555057445535022590113992878808414103915815\ 343452290752899160068897221939949134288017374302038030819697518140766838\ 203376306320579613848510557287486338333772004379201766944394501760443905\ 768452875855824491861492337741580908349098121613445207432226300814244017\ 726101690530002297500950495777344224937859095667008712229151357550485065\ 754795885021267996415627826408093988263774212542740665637056672090577876\ 029896836739699184042510249164044099397064917285108839953679204015733097\ 803082414734480816879401320405056315338077032175492527818867074206208957\ 889582074107961590323200841011820464731739685777295474100303379332590302\ 899763860421626373413307644422641129457422877293814813913740604499270139\ 097468551664459176092688224978827331591287286627574144160544298786726313\ 635784751963083917641651515397487006311570718862466939435711129979916242\ 616475542141527070307831939057943941215224070094860239309320613717559309\ 291525608124744972721740436460533556952024439579627239605174747184529503\ 526024049537221786218263558903339381034652843420440330482002264888876822\ 221224952434306080619618126238419329070360715202699722924673479138637194\ 474802297507321653388765615552719681872545208382336240819831616538279707\ 606901662726506390305413732459323273020771131763426313435639641697490096\ 765428242056113679790807476106504986341363705868511535470779357134548471\ 243416766173756445172649590847872068662476630993244478697376723009634080\ 528749134093671278497803158083480338701651503572836361265165094415064441\ 633838578610068986527462674712882499454705772575658520240960542561228160\ 907958735011982677232257699682436751856650238424681164493817259531387371\ 170906825580712665764122461743059271459040932019803688036714354378593513\ -855 59633602686773876003920 10 The smallest empirical delta from, 100, to , 200, is 0.1303065704 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 3 n 3 n trunc(---) trunc(---) 4 4 A(n) d(3 n) 9 B(n) d(3 n) 9 Lemma: , -----------------------, and , -----------------------, n n 34 34 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1190 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 34 (-34 + 1190 ) (-35 + 1190 )/ 3 n trunc(---) 4 B(n) d(3 n) 9 But , B1(n) = -----------------------, hence n 34 1/2 (1/4) n B1(n), is of the order , (34/9 exp(3) (21 + 12 3 ) 9 ) 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 1/4 1/2 34 9 ln(2346 + 68 1190 ) + 3 - ln(-------) 9 That in floating-point is, 0.029167545 It follows that an irrationality measure for c is 1/2 4 ln(69 + 2 1190 ) - ----------------------------------- 1/2 -2 ln(69 + 2 1190 ) + 6 + 3 ln(3) that equals, approximately 35.28468183 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 15, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 35 1/2 2/3 (1/3) 1/2 3 35 that happens to be equal to, -1/3 35 3 arctan(------------) (2/3) 35 - 70 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 35 ln(1 + 35 - 35 ) + 1/3 35 ln(1 + 35 ), alias, 0.99297147867008281213 Then c is irrational, and has irrationality measure at most, 1/2 2 ln(71 + 12 35 ) ---------------------, that equals, 5.0678336983339678796 1/2 ln(71 + 12 35 ) - 3 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (213 n - 355 n + 95) E(n - 1) E(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1225 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1645 c - 4900/3 and in Maple format E(n) = 35*(6*n-5)*(213*n^2-355*n+95)/(3*n-4)/(3*n-2)/n*E(n-1)-1225*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1645*c-4900/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 35 (6 n - 5) (213 n - 355 n + 95) B(n - 1) B(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1225 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1645 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 35*(6*n-5)*(213*n^2-355*n+95)/(3*n-4)/(3*n-2)/n*B(n-1)-1225*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1645 and 2 35 (6 n - 5) (213 n - 355 n + 95) A(n - 1) A(n) = ------------------------------------------- (3 n - 4) (3 n - 2) n 1225 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -4900/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 35*(6*n-5)*(213*n^2-355*n+95)/(3*n-4)/(3*n-2)/n*A(n-1)-1225*(3*n-1)*(3*n -5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -4900/3 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 9683469929075115674125557684561127072434036940651024942129065043848449080334\ 723095062739757947912052831317248806270619531834154559147322952983105926\ 462470162598076556177043250054689356415044641988820590359631555372687322\ 482364220572842925152384915195935276105918707741245103838531376611942739\ 027875775318789317873206073721705781049857983900765476390163431127345336\ 673079349961717203215469740461171617400078321849809618378591588954899258\ 813833899829069120568488157749456476263495447346799229927836296085498273\ 574878232619007958669087636516262789618599911055483285613536967395837576\ / 73639799346525067755613473725975 / 97520121545127199974767823655367897\ / 243567664209620486663139522563833225266575912296499840671734303714680845\ 669598445361394707379094922276813924936199842240299863004872687741658407\ 628391467808444232428846004582803933862186687135880607525665992663615297\ 239498125353169144279427492745270666720920733575644934999826031819056750\ 952688986809093376730512187831959861170459677611135516183334531879083488\ 124592309831885624676839806629538858271781608900568609484278106250699066\ 625710170443513909012514213770044097772003189845027994964107773950277581\ 820502806336352166744417329140483247726824164114744705694917519719631590\ 4 and its differene from c is 0.16276099748673528402072518422762961276182944360144401743991708006924846904\ 140958473834487920507471186576249170782188967567374155256206787195246424\ 172988893961029903743953406705453254950589185003008863107051203183977059\ 059537024902371291244535301115411586761630595818390222309901668756988927\ 360471629128173323867394852493114954801145778867164120026292230543206329\ 576387691535216676519997654341992660623753034937561874080037895178246191\ 071402311745238401708832210186418521969634731825814464851734854187178287\ 069559648727950465147479519336928041264292376742326921148326386662804227\ 737408596785293092184481257067721589249959778910495545157989105248745622\ 892199085858958048678139298238253225935076514115920701195852515477408708\ 764855463528580733336191136187887216969935488743247959693440336319471864\ 579329361936324673125204177150390694488446546111904284516650901334232639\ 810579343187043158065158798809328510633592079282123872149951019873190218\ 316319528549939819680293652142149936903130745172737023159730315684959546\ 601621241832160992246683756669621710123123062083319557521489383636914422\ 782870963766407373812673585263854111998174815204954636293249217096338704\ 219344633839609163832880498721550714100087174519218527178524316331417386\ 738521962023359228132641019736819059124231090396029235021913226779165217\ 438140569208524368810085980426462511521405474177445625356119483097054665\ 875852285170965093255898646141271554626592683811436211129682754906681275\ 608548391365895320574601014891671318976354246379629043705654741734467933\ 827269363323792830784877828368083312953683442485037129911558168521156745\ 305262557291216963238782097881854402003536644710471972713120804978477967\ 920349261511258707392806281634686512789919008780858457709437132213362253\ 298248108298295048680430228809817756065540689340285099455935424923400307\ 442376509730589761471614472600497671392560822353683510796322484826742875\ 359847099740014466026872437256817208499095474104095844212324004174208053\ 980976126480038829289368681249989480601520374826701695435688420062035392\ 372550092351180705393567271210892445250045113338902736834601314748425673\ 278257594381861480059185720123594734217993257781633327181675815991102237\ 156083932359503820556597322631099590197146970865679698654064265566671938\ 769100852384810843298801440303253846122029978284868943726104610626133313\ 495316658465466383995997965605489003084311625917500836253317363387838343\ 130739850017391996866967325505219786198771153688697471815765626628307592\ 295688856297475771111927107106534625161347828699801912263125640421158633\ 008696639163568323323226928988032256443713912352354529366384349646217416\ 580516717556280347176823026574541020358433302064930123969580210445441742\ 857596352430331827145940214128806891125944711488386413089659662111434782\ 397293862456247578139171776581632234515198599966882114452483013620931604\ 427466430437936612905109615298345869006065013246129872603637998079795905\ 842251157114382061521075222660461214212018970632176092073850808127743812\ 834983353257087268634962985217353215715294981238403874261596512557507649\ 942402763024157899062058728954396929920617127154383971820482667588821063\ 184773490702954787228211945361321434518093132969398602394752616729339792\ 660701803561299885369121897075021523142140830974633823132050322410445163\ 240912711041277912296104465972767602584932103279450735733585508716167609\ 724024631089728932483148618894829532562873219673100432526620863599436760\ 173102299775862739142689541103658623135244397772956526318997667375110614\ 031628768925543172237506806213673375901590915001732860963623893890895736\ 463249935790867509208921276598405267452076851526044644258418478983307493\ 108973586456432015835151828916611285529651069160922673185499213077268122\ 496531556262903291315442761059807906095343083799092139614392786326971928\ 091120802796632343978431141535952866769179678791201005476490218909749388\ 247211884194859655051833282065000340489887435612467622625822314450410577\ 188227464975742682980690248194589249820379671163395093111116500878734095\ 046348823620331750637335662216609515290728596834497758981951271874414436\ 015965133402111088928278323771209717756550682698857423183736508741276813\ 850142364171950960172768485443979570624509116072570567420065345472536329\ 246497181497083674303866844601084460535672045197478175190790357307651526\ 498757079543598434165146049496414089916302509739567647702113755859863363\ 110327128685984362914172391405290820641632680594122137846277797103003789\ 259902080059335857562684803082839380613667649046436343989820135178181475\ 209218765127467322161798453346260948867307182127233714465672423215233181\ 639021333357267008600184005437665574719186577313700665687230379634980456\ 245490749607669099361168585822860643514443530250000126380031507139839282\ 430384096036376851663806938642255066119954723459923745093735999071797587\ 654678661879633700059434497767468857872287807066139304505299902950910934\ 324761800647962874036273251779644063160618178849242429392539652148637801\ 686975157263226981910209861057169457479941810038342111910736683867788116\ 139702033471463939223704059782765331718397414670397220469451511412711414\ 961867499927221316907462254107603280182673468015518660143613157742805055\ 180048553573043123449264824423450660092869824959742871792628049411253748\ 482563398555187998379684703014509218520179472389148671319633847182587136\ 321612099201394984227507069726895012151466062611337661050176959371980575\ 798796387807133989800996327713274163644010586621453148743050269828296031\ 799885848385111182720505320026284479497494436125586204414940840530360806\ 936785094047123521340603504923206493761368276413493494411165069031692622\ 591607713213189945400290029513722829687165761551131669630527758838173005\ 884051576462622238610841905660859196375436143602920135488628726645461159\ 097967438602397883685320181161099441041439094935805837585589006977828361\ 998351557035019192879451566854231185876605931168043282011620182961202754\ 337875801944253711431960074987224608274038135873991017949388356903813220\ 198414931637843143056740484432253739366345375204105746621102863235653619\ 012286209922346573694501543229980963058376165113067712869666071315218229\ 324815548389835823090825357513095385585759538868915656612804900686962241\ -860 280497409345289140 10 The smallest empirical delta from, 100, to , 200, is 0.3861767239 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(3 n) B(n) d(3 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 (21 + 12 3 )| |--------------------| | 1/2 | \-14700 + 2485 35 / Hence | A1(n) | C | c - ----- | <= ----------------------- | B1(n) | / 1/2 1/2 \n |35 (21 + 12 3 )| |--------------------| | 1/2 | \-14700 + 2485 35 / B(n) d(3 n) But , B1(n) = -----------, hence n 35 1/2 n B1(n), is of the order , (35 exp(3) (21 + 12 3 )) 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 ) + 3 - ln(35) That in floating-point is, 0.245831179 It follows that an irrationality measure for c is 1/2 2 ln(71 + 12 35 ) --------------------- 1/2 ln(71 + 12 35 ) - 3 that equals, approximately 5.067832258 ------------------------------------------ 1 / | 1 We are unable, with this method to prove the irrationality of, | -------- dx, | 3 | x / 1 + ---- 0 36 but it seems that a sharperning of the divisibility lemma might lead to a proof since the smallest empirical delta from, 100, to , 200, is positive:, 0.1310444515 ----------------------------------------------- 1 / | 1 Proposition Number, 16, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 37 1/2 2/3 (1/3) 1/2 3 37 that happens to be equal to, -1/3 37 3 arctan(------------) (2/3) 37 - 74 (1/3) (1/3) (1/3) (2/3) (1/3) + 1/3 37 ln(1 + 37 ) - 1/6 37 ln(1 + 37 - 37 ), alias, 0.9933456606449648160 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(75 + 2 1406 ) - -----------------------------------, that equals, 27.631496952074755211 1/2 -2 ln(75 + 2 1406 ) + 6 + 3 ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (675 n - 1125 n + 301) E(n - 1) E(n) = 37/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1369 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions 5513 c E(0) = c, E(1) = ------ - 5476/3 3 and in Maple format E(n) = 37/3*(6*n-5)*(675*n^2-1125*n+301)/(3*n-4)/(3*n-2)/n*E(n-1)-1369*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 5513/3*c-5476/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 (6 n - 5) (675 n - 1125 n + 301) B(n - 1) B(n) = 37/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1369 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 5513/3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 37/3*(6*n-5)*(675*n^2-1125*n+301)/(3*n-4)/(3*n-2)/n*B(n-1)-1369*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 5513/3 and 2 (6 n - 5) (675 n - 1125 n + 301) A(n - 1) A(n) = 37/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1369 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -5476/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 37/3*(6*n-5)*(675*n^2-1125*n+301)/(3*n-4)/(3*n-2)/n*A(n-1)-1369*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -5476/3 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 1705985766006267985652240640074340873198374454797357253560795416569256447618\ 045704897606367822799954166976513225848433650296315493047773370117301803\ 014249684002454460909566162884745393275421105928903536705197307134748604\ 041925714415630592930877499437845264272630100632679246200611460846124583\ 994812055685276506724745753440030248697209171235522363670113178947147853\ 274561961678145884459376672226478366098620025825126077184587403638688395\ 236846223212259458288541447926234667401885795298241979253501924400571578\ 868686911421045825781634464180576270717006940317261504875930389345863916\ 682228180647099180054894391058503369819673964194619203292962361852599590\ 868301558534435819292521694320471194151302039318108439427417626081140904\ / 9307018687788511374695182239353 / 171741402171988788591485829393924583\ / 582548576394229632050323165432187557646095444301524392385454671193166165\ 934055268627622798996635536415301038012490341337999665430826654407894651\ 815739728594710237931439807599326006136306982270405565436112981602537579\ 865729223120082198827715312945001205032151142246420058335996077276277924\ 568227585348477160939313588466094451881332268942869242714235981958090446\ 825566530967407174828027777174046979792675158066054879461053692992342257\ 966903544115371356455912428000158972044947722985334602594599161164610402\ 980893988106577011970044560525263217413479641464480194970147279014420033\ 843731747200354281767116732937598416555308534202194368365816846319208889\ 16647886875573642708636447047297546157315555762526865744227357595116800 and its differene from c is 0.48932684420065067484985778328336261634123923264886219033559882943940736127\ 537074719143369649173361742556889458456965138026782091920814638691246616\ 828326143784190038348610494431905469405196534502154600820775254916327676\ 789340382127727208672246012134217522351980632562602686576460200723770147\ 927773000304116103659413552399011026310931716183003124284107087483186672\ 174232459515773947247567090196863558822291683151618737845422261022617865\ 120326326020033744299190376803762503327957221688692548148990289993234118\ 034749845877097690518861968540568611913631511278652156667977628240377476\ 290898860886283870240814641422154373515885127866387543262790275299046952\ 979919677160414534616109373357652211514463991278953522054253493539591881\ 691088275492902609378181640607172329384920963243530395057658925985831331\ 520012780520963291713292629105708586984230230912954628598039809802516587\ 266579057215287713884111212381020246737568134948762493844759733502685651\ 087188604387104871970128428661574664941144281955501402913962945621134350\ 430660590800702785279406504506908408737195055978962873806057246737490114\ 960806382797500713404021639441564080180712018668190370844869995503032345\ 786897839846852842223480635642261305149588153092457303399387868071885503\ 809660067295309716926504790363217427697431828808970074917898292102654470\ 049985356185151333591159884976297963070004003923013115061230278184066203\ 190067211749218304030030271458468544590378783560342753020042682440722970\ 425529124286754134338485855373178177217261023919728199734667545346869914\ 223178168056716408758733174380030369489068657177334002589258948462860711\ 596889307924121333412529450811035486829206308499576529680780992949522847\ 231456660617415655416972781967265298445298736074067801967535948743461383\ 926795184006802446025598462892273562413428305933319171357818769743820547\ 464960205750626973836426333092088456851410734265813999578019803023214144\ 821018885335818978947651817007625613003897758660252489408504857951758485\ 581264213469382149382287725586370757866991315344565985567999732820630867\ 263665591934691718120727311274436664325292278691146157711994053586815782\ 844762668041715052571446568923659936387075096563184002897072600357247934\ 070767133263390233646375932835103067092360525996215798519915275834833303\ 373271627737453449299211809423884615893879162629702638382583753703164972\ 881428347402396415616406204414923402216290403884864346126054067733272929\ 809924748679812806989984373809873273208175615694658559443288059048604146\ 359276330365469309564850288481753043014266819128898045286962716616970449\ 090432642294943660924564074450095793661746937373666964806907066493359707\ 715982672974572505163542792062712014908099620950472767897987298345037185\ 533631573289630556366162275693271653770799816367931410693739880913596741\ 605589406660444060938057812427732396977779989823717065695328278665422878\ 974899586596228154645168600530947216573124314381208169430952159907766595\ 103278216249613479330295691028794673665226612885997849304878323951371648\ 305961604872042555784897098098528628137373676346123521322075097553407272\ 699185910497942194975444268949675021576083993253153475929458224566591665\ 649586918483485376938190559963720231711195882918561051887243144384056481\ 321711323811681770870049597985999383608102999301137835332659220703943730\ 199839557612159343484856333099689181455006980445449579714665298002264584\ 567954407560639815256590090304208582333834979749977590866348762709230072\ 203460047966921423754339494664716871682601373430686325208722332671859359\ 188473433172143238918287662226728297137057611587841595964150299140704456\ 344253927145007786541472655927923170826305981028177699873028172388265723\ 030709186841799692702764486357049323234862121522326648096830143925691538\ 545653372238982565274670746590851454948475190541489873743807360574116514\ 931632075287534187490395054801058289674290118365100162668744331423918329\ 369201213518827660918084852615521556418565077932914979226849687848776336\ 174155895297340123834280532969713323627662697928202464870242494986717010\ 648363936254945843949491846172883684532108524004819762027305247224707735\ 817991346421312095253655442001359667458776146372413120175157272909351208\ 303944469900750611376774997604420042595284948094134137265134362873219640\ 743912657275984498094004425412505949151594630416002957973058436753479010\ 955637423582054456931194223736848506989296844680280895787379665950469583\ 021264384175597211224376581113083131959623946505627641025550192369291350\ 094404567775391255877355193696800525055129081067757521863597980338345218\ 746819236537743443475793529178362170362645958260709722283571548788079143\ 282667496239421516978171874335539359945853375671249020333904101126128804\ 719177073623291027533982023865323497267446295575963600318408141892448844\ 875015086272256004439492105399536058662175794651302516088344912620414059\ 338280987074282907237777812328908030469332798085366851905777481238282897\ 552526525679072160864808829186841157000933383608477737350732033876195272\ 991756280823001163629825534769179949229016649588614451409265416664365677\ 148641793642677393892750221037388786512544715562145764995109286749570644\ 393555618775842325988529501468918685621208091608979931230845540415814223\ 056814079973012265411067470189872582980061036338500128177754520237022031\ 173821262735779396130871748934468873099550268196395081511150454294744920\ 078487897420206110363482724548106922450817638463626299991603910534059514\ 337314433105747815320532255089351303521124013036801477165620461792378047\ 024114516602455349795234847797473795717890794482533587770129618629063470\ 740717416828502287005210921794939675928643920131640974727680081800422482\ 175712851884650462932562859464164725494365854518810901318565649267669014\ 848197323134371332222402243916396823747027210898032807406464831595888548\ 159677232545023566384044120633035859429346825197779165247133575012190886\ 853813800655545639763558925218363430609355127358588972070703848662068472\ 901716458594757742419094059022882591438788781185839842789899103173497033\ 093157069214772563357513215385465902279145364039363472517211138661190906\ 662433238214912965162458626716990411556761232696856437072674699296597189\ 976594424353201555166633647416828801510352781876598286865798277823792585\ -870 21269464 10 The smallest empirical delta from, 100, to , 200, is 0.1360684052 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 3 n 3 n trunc(---) trunc(---) 4 4 A(n) d(3 n) 9 B(n) d(3 n) 9 Lemma: , -----------------------, and , -----------------------, n n 37 37 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1406 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 37 (-37 + 1406 ) (-38 + 1406 )/ 3 n trunc(---) 4 B(n) d(3 n) 9 But , B1(n) = -----------------------, hence n 37 1/2 (1/4) n B1(n), is of the order , (37/9 exp(3) (21 + 12 3 ) 9 ) 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 1/4 1/2 37 9 ln(2775 + 74 1406 ) + 3 - ln(-------) 9 That in floating-point is, 0.037549522 It follows that an irrationality measure for c is 1/2 4 ln(75 + 2 1406 ) - ----------------------------------- 1/2 -2 ln(75 + 2 1406 ) + 6 + 3 ln(3) that equals, approximately 27.63149747 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 17, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 38 1/2 2/3 (1/3) 1/2 3 38 that happens to be equal to, -1/3 38 3 arctan(------------) (2/3) 38 - 76 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 38 ln(1 + 38 - 38 ) + 1/3 38 ln(1 + 38 ), alias, 0.99351819786731638893 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(77 + 2 1482 ) - ---------------------------------, that equals, 6.7718417709659067560 1/2 -2 ln(77 + 2 1482 ) + 6 + ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 38 (6 n - 5) (231 n - 385 n + 103) E(n - 1) E(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 1444 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 1938 c - 5776/3 and in Maple format E(n) = 38*(6*n-5)*(231*n^2-385*n+103)/(3*n-4)/(3*n-2)/n*E(n-1)-1444*(3*n-1)*(3* n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 1938*c-5776/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 38 (6 n - 5) (231 n - 385 n + 103) B(n - 1) B(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 1444 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 1938 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 38*(6*n-5)*(231*n^2-385*n+103)/(3*n-4)/(3*n-2)/n*B(n-1)-1444*(3*n-1)*(3* n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 1938 and 2 38 (6 n - 5) (231 n - 385 n + 103) A(n - 1) A(n) = -------------------------------------------- (3 n - 4) (3 n - 2) n 1444 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -5776/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 38*(6*n-5)*(231*n^2-385*n+103)/(3*n-4)/(3*n-2)/n*A(n-1)-1444*(3*n-1)*(3* n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -5776/3 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 7514083717400931164080541201925567963430900548462384265194958167665715818337\ 032082632230725333806142342391960148972206210616436250879042752259561650\ 285750188576707803607761485916325312483316597733475071794310548197854240\ 236353893793850576403156098752820069720748574621117209750215944437291364\ 215164428945188081082168732353300466168654620914039470524228772616274543\ 047743301103881819767298428292081971909544358824262662116823837045626180\ 659382398591793888353490292422861496966884437418369069850497561005059167\ 095089823486441354931142338021129357490413203946030681437044378170974710\ 798315631427580025000710887271775323995628031800842356660364287098129547\ / 70045041501553 / 75631062757890534532692215269277284962115140178561897\ / 796259150600819993126747145976700345314710578705714772084702987978239403\ 905925456428281497489868539120030851871052133339696990221774621679966034\ 009726273632809079798492543384957676757515890549403288618740467464235427\ 101992287813535521916963997428613975746638270801181664736669121038379890\ 075457477047795907545317304052274176820178406161169851983873890424949267\ 216155597392839762993265737459002749165852599326522076913712462772236214\ 247708837804987623678583561932187933691013485229034708417928198822351168\ 135686865949970291744911062334152367965393044923049324791679204648841364\ 8974520053350689531772041638343953800 and its differene from c is 0.13105956239322478697787657492512078750299063213575257563315121424619903909\ 749764285476210070083628380748735494414687616331246808438381492392546992\ 106201630973625448255073505660067782430921687002549432534068549310413977\ 727480718665253617370071263888767296645915052970533037171993378595486615\ 586544578675899877627622459537329561763429656770013587007027195193023527\ 779176303521436571870515195482846267320481008619812078351520114271942337\ 549203075334537116893553005422007431488040380957127239866210890362551753\ 538982688351840669425679242520099077414984565302340059964096733908192339\ 445440288587155888920064324194873778187222910186978911329517039226173787\ 115392664316025379358007981991108355140297087320903006320744656946949753\ 805223628149163479241557470644246424373056301668264023895214685349865592\ 350904452675886369855854079075661007643469511783672350054588319006935187\ 520367040643054909963062830011922097814515966055529983427473165352593963\ 800977062646290198857554987660720100164566462601719202270059387176272875\ 600742803010718057366358565131186146807157739681915502933050293084506051\ 900393790509381658246838226172564382983061314167216201305317952022117481\ 897252468135310442474092376892135723233363638655911278698086264782597769\ 867048163811060801752774015332100993120368209372328149143578857024631523\ 899628084949029056775554449619718191781295595415475456124400719742421803\ 748890571529430855582248336691731256893314507601703503520023065858384845\ 204388460411165608635169156613891376853552881251561460480892995824668845\ 607340638945525354343470100711982555735997454180663220950180735420090161\ 005104122287819403762939795300506379625026136164937577754169841012229157\ 939333523596359535903197291104936123588680191639390585461357484511333158\ 546244966589870713663264502610156635011768056815585708192569673206141076\ 693504557173299089054933929054935172939944175552513290833421972152632434\ 651172885587431159406176949696264782046192957747386439674989082459106011\ 432818591139921153079145485871335207063112038675684334842879644005712754\ 234850635751110399033498158479603048956010123644169616529056641788882702\ 338553494765127157157256406673162038835172243063960777584660086631967820\ 629768346765391058724076148521515966023279685194038708447728820487291656\ 420966011343134821511544160220080649623652572036254781727573842389685366\ 027271801652624936489123969881695800499607661968291912897338418926386594\ 377086498190986305278631401618412235782355168068889929026390040482516589\ 148967308015524447818377038614825463116031581431544379932532671756235516\ 183280923741867598581643594546148291399923090691118426333823194300317097\ 260184843737119829915430043390539654243817418790260470646638870859601649\ 962004244946022103981945757261186378724384888096540487083199284284711596\ 508990897647794567655041525622683696555263266352875291148714914890765953\ 118617742436380795616692351373711943601151903502555797536715626991984258\ 370813972509626495929624396958655880383128380267151390736440386852561906\ 384676368216522647838165215103850476689863422590635906842176600881073939\ 125217687395172338934217166133445977356703466974211794676622862624649137\ 067958214964345043813022048503101899056255035818110382331277392126602072\ 677242409926980200969178890874880324752291378715707830157010116183987654\ 130934738112772827079652739111075313890404919254694293559362920603213184\ 225024531066974973646953486637837015314451564019144235199050607280421360\ 921058906746026797979238086435502773135751341127873385116224780985484230\ 384336970566679264925155826189424484593049231553679644432260972364539426\ 919264760205369485470522648945133724990747621294157416524158359181852105\ 967395325295538560983788367455148177952459259282946826500953270070400041\ 341726691625887716770317631064011421522216746357579301644917882709822706\ 992130967586176231243008900382101987520581357793235716797509527182198798\ 996275764606735988405277702388248897760659935214524952896701214269664753\ 347927125054143028203216520540820754008157389148708954965188207109129271\ 151376544871940773384354816712223926393858291618246481595657142021588812\ 127311308852675073025669649003689375555547453095623669103669621038759401\ 054089710724706316490725233668246004943452426360369358721204294612119319\ 858142650458471629649834977677654544954547302122982207980235901670152773\ 351906322925599934569311534396926883606319677930451142080549685784020471\ 604565099988258041309605499157869348284202048930301685233046698710234139\ 492977823053413613494287351028327654102104789797333075863770092090227353\ 152652059652915103879096057162746619027219140200584076522168829728700549\ 028778436389027137189839126498038130583149922933403263075191031480577240\ 846053743788674592980663193190043276658589508319391846691249491756187790\ 068305113680051673370162274536062401695670624616086373325136306209963835\ 584007369816870392498628539581902407258785526682424008116698582405071973\ 649036860159000666908037803000762464577154007184078127204768316687543016\ 819984818882307035471579108002828133538867739475763405211166946898297774\ 502067588882293716994335975336925370275686234539431426020908478197325983\ 389223265869974283221235951922831238696618800817449184123755495007271939\ 028427640557693051081011926779654941278582406781873687648774746321147462\ 315994142279602148732900932262095919197427290040842005674234269184178556\ 816423961546984841006417579270376586977085720995692705317074654775932975\ 299959592517446468877020697195892444039157310906129500425386192157533031\ 538769029332402766393849730170096895167661911619498524282668518417442777\ 708681478429386990863047838516918685249891848621589296939131347058838629\ 928191112253465253906247365091852387842129375340828724151841123797851415\ 603442240774802768277819717010437305265842946391992223858155793645549196\ 465556136958325774563240093507486037675806523462516977458259171973903993\ 012967865487923564736896309811175348406335270242507509595842170844568782\ 101953805460071420480462487258918275526645716535306207634119724464353593\ 441883191416802534195628329410207176378465491816685729775229368834649902\ 800529439498103716962718963044823002366290265565545935360608657042210570\ 149234665507479841174834238404570438421291553139045472425578848605425729\ -874 0454 10 The smallest empirical delta from, 100, to , 200, is 0.3007294741 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 38 38 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ Hence | A1(n) | C | c - ----- | <= --------------------------------------- | B1(n) | / 1/2 1/2 \n | 1482 (21 + 12 3 ) | |- ----------------------------------| | 1/2 1/2 | \ 38 (-38 + 1482 ) (-39 + 1482 )/ trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 38 1/2 1/2 n B1(n), is of the order , (38/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 38 3 ln(2926 + 76 1482 ) + 3 - ln(-------) 3 That in floating-point is, 0.173254924 It follows that an irrationality measure for c is 1/2 4 ln(77 + 2 1482 ) - --------------------------------- 1/2 -2 ln(77 + 2 1482 ) + 6 + ln(3) that equals, approximately 6.771841728 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 18, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 39 1/2 2/3 (1/3) 1/2 3 39 that happens to be equal to, -1/3 39 3 arctan(------------) (2/3) 39 - 78 (1/3) (2/3) (1/3) (1/3) (1/3) - 1/6 39 ln(1 + 39 - 39 ) + 1/3 39 ln(1 + 39 ), alias, 0.99368201353250523296 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(79 + 4 390 ) - ---------------------------------, that equals, 24.419241843641173377 1/2 3 ln(3) - 2 ln(79 + 4 390 ) + 6 Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 13 (6 n - 5) (711 n - 1185 n + 317) E(n - 1) E(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1521 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions E(0) = c, E(1) = 2041 c - 2028 and in Maple format E(n) = 13*(6*n-5)*(711*n^2-1185*n+317)/(3*n-4)/(3*n-2)/n*E(n-1)-1521*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 2041*c-2028 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 13 (6 n - 5) (711 n - 1185 n + 317) B(n - 1) B(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1521 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 2041 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 13*(6*n-5)*(711*n^2-1185*n+317)/(3*n-4)/(3*n-2)/n*B(n-1)-1521*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 2041 and 2 13 (6 n - 5) (711 n - 1185 n + 317) A(n - 1) A(n) = --------------------------------------------- (3 n - 4) (3 n - 2) n 1521 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -2028 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 13*(6*n-5)*(711*n^2-1185*n+317)/(3*n-4)/(3*n-2)/n*A(n-1)-1521*(3*n-1)*(3 *n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -2028 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 3912481496814000011155587745990817571306966209103890665658877330685577379616\ 277787918767667801969345804865084430157172647989150307692944339293488193\ 448085787996683887489431798602756455178480812586796291334182073756169994\ 959785271784741665276167537931850074436050271079145498742680188835868584\ 897935750110175796762669647095811154321160304714334371092073371356065409\ 570557252290537776510827112822538274687526487814585461016718474406719059\ 808356611380737782966020495140559334014934785432935203796424782512478308\ 732495564206007895104205768746015016108437157843549775902721709510932890\ 987689568214392542147798271046513106676711322201531212919081691579844995\ 055329814452930493250093645675426219945905416905965032339894149826072002\ / 209739005629849509001813647599400243451 / 3937357669286237194013261998\ / 031678861905200950611036685920458381689055558039652349258512976825804160\ 314011244948447956950076405651312090848304034470437663617990219033155351\ 409533541460924679892546225422077873868436010052351036811454551291244227\ 552051491031479717364194110428412355806688195657217301445531771517352880\ 370122937165176278848263778367989961773345106634543223328155329107484929\ 710098392169790030285962532680967639302268064665227778239851435134150840\ 693817294903471755037935428252602709973134041707484624091210123148492241\ 405428269039627343433441214769104125159229912015683798732348251544067099\ 872114162490480853793409324956799491563063217897802182890783490537017503\ 912097396005127575611134921145669619042915704660117556149465592939572293\ 562201197958400 and its differene from c is 0.45983686449861560531978820303126323142809580363600073894484743323851295992\ 728664620345779770353711300925308922661011217108762005146064692274971291\ 177055028446124900283448660808499144943813246059281895889752114821188141\ 214102566940992582085689405592153451225946784421529389601221962312480752\ 187428382061822745621038823244701529110580471719275155876155993631062001\ 777575581100657752926119611711452778500111102293037272185456154505223263\ 939816873966783387557473349457277830975987187382597765944294845699290865\ 588595304000144174137773751353381691687237883295749968968342400253395196\ 846885614625580654050846456946471376442962406486998617219317574231355305\ 541608722208678163903416044720339132398807578501711660616764112087610333\ 114371591874008963147641993049216326907154611785457753366214490261690942\ 447715648064357519867636832302609498643414401729221159746875676079865892\ 255252997332506019227778172655147982548479812484902789699415236028242966\ 248272912042037900083373871519909689209504248065208028127548994079423165\ 062843646233183735053570101079156036592033787697003938219026024753770923\ 304417724811247424947126790566550442734143748510125381271386851796727120\ 649263221965991793674353850942936206494560233122921904429822573757742752\ 013436853453265202192407065170719952957351820060514639284434631633528114\ 091095122831320964545936122003628903055843181650370116326207289256183468\ 199974616846802974245380550450582880682301008302824332693900706864649937\ 191658164310414529174021413214395298900559586563372916637958969819236974\ 323671085223415427736582560241365871630718725614699435854455387718793320\ 595587696916763436184656032668042559276076299820746008777639447979049771\ 853349138642455131930709021029759432263336784497519404848692108780731224\ 746716374522698244093392106146078571112642436590300930384175784698875088\ 220354578037027486117719281424698193266332930312610966103898209793922752\ 018183029252079394071515470831153574114891039586413905506552072807123614\ 693424581497765340190508668379021350899891865847114822613657655704697968\ 718293520181411410988129172908072074894144112151957699705512496182581225\ 021372352990168619488519825024149256587265373733075252655094396707866596\ 483021531343076731611239187340556454225196007620323973194108545948664833\ 254534207568078232848208969806965976947271007674214580539087177839417766\ 875490405985550849167949753413238397416268602230940373722689998367815305\ 033724182915959653743664867754984078565040595055325956232175510412121129\ 232556906806746030703168820572792678112661193644337879879921435001670564\ 913536088491059929119574479308319064681281452834285208826739160259155350\ 085767295822110106830692027478339342651508084625738059401242223220296290\ 515730422049922484610446765993112341437283454152528841388102578538189541\ 692629492177718836204724610381266298081450822872315677885666019348746628\ 964806303369322862233256152594267544394351014286514738826311102089362197\ 028356126318642922593054544298395032723624713900851984479820308997472032\ 733324908634721148228157257720613549785025215218616198435788353024512446\ 126456359504787116932020591175742833249692660133328398330535938409126810\ 189716030811944638189346254213911830041479189019080737442626211743100625\ 843745547846713966816488508895968347415413692077100803936724171810595195\ 611330790992174187287392614509346578077360357739243930213222566976571877\ 711474795266632159333654404815261432988819421315818648662722963160895596\ 153124437934689909377568135941448485822402095273422870548951886158677458\ 963762452223669132625071730349903942855736259752952755473565306351416908\ 460671069469924388250063813662191023058298193172139358507107008670488215\ 117729081230777669629758130778808217615565026802790239464327725910577972\ 503719229821793022055115059513106808898260811553240816998753535931034798\ 918123634042801085548695064998451042340219878145730475572425546569348179\ 244729982566728733381188049774815517912433535141950269400080343124905737\ 783577744387694073922219348757663572426783820931458807260976740035239527\ 407567177907248986613031820223970030932198836765383396000683890074225329\ 709156869132123047183476268486433690140187041028754172519249897291810785\ 319291166874763036747833554535336355115917369250339657217576823220935397\ 667855749042278542514492291072394872650416047445438030987327472314997279\ 084670887064670688301545876531957088993731388868136625350522569131680373\ 456792257965108759883042541741102990206066160325287639885605403351301455\ 138022565167087799996764532602904790934123253452365885584131531368074299\ 646348150417771748371631449972958665238154802793776862683403487405677011\ 600578083736507937746477477420843326377676639976271712884470229989334695\ 375971212257149296483466232650817827818001703285405153941601784462353345\ 842194088810879553109798355722696988055093267094639255796384049962305640\ 484500809112692674378864651291596243172791072416682638648155606055971316\ 090129862962503914348288659515560780097038272964483467453160873188971149\ 034266267031652272328421415210402889821287790756923010036920023233814114\ 161315965512510443312708527693352697300316194493230379136291111229736518\ 797310321392549545501575886764273860787409848860648549017817403069122772\ 123107489218463981668214495219227442764681630507053591727615438283198412\ 042546018183385796008230250053332035465799513542367715198812909540568983\ 588391299079074958900083360567117521882254266599011092824703290540669386\ 607917368039053314325059386941678043712237898262726263994518793875448677\ 061031446653980624283825803117395088826862699401260816282687998856586392\ 547149347476723152585904916385324337970168247798698297333972133358782324\ 156214415094929858698979294919030282402458431432714586874097911865625617\ 878619475299702168595888397351793891936728951915904352181290888265778297\ 301724434500764365695062159241592232960164192005924744103726949784332780\ 321477673529356593644723080569797017616348750829132108199548037912990686\ 294953912869631650750462112414710806857844652071274931579029337889193280\ 209124375628676454113065656791150239675816423974433562452584262809099941\ 863612436839026476670382802484876942843839313245626617282988252543095651\ 30105682560218444991872142804001852284516933761859866009397702728415324 -879 10 The smallest empirical delta from, 100, to , 200, is 0.1416066702 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 trunc(n/2) trunc(n/2) A(n) d(3 n) 3 B(n) d(3 n) 3 Lemma: , -----------------------, and , -----------------------, n n 13 13 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |---------------------| | 1/2 | \-60840 + 3081 390 / Hence | A1(n) | C | c - ----- | <= ------------------------ | B1(n) | / 1/2 1/2 \n |390 (21 + 12 3 )| |---------------------| | 1/2 | \-60840 + 3081 390 / trunc(n/2) B(n) d(3 n) 3 But , B1(n) = -----------------------, hence n 13 1/2 1/2 n B1(n), is of the order , (13/3 exp(3) (21 + 12 3 ) 3 ) 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 1/2 13 3 ln(3081 + 156 390 ) + 3 - ln(-------) 3 That in floating-point is, 0.042699819 It follows that an irrationality measure for c is 1/2 4 ln(79 + 4 390 ) - --------------------------------- 1/2 3 ln(3) - 2 ln(79 + 4 390 ) + 6 that equals, approximately 24.41930302 ------------------------------------------ ----------------------------------------------- 1 / | 1 Proposition Number, 19, : Let c be the constant, | -------- dx, | 3 | x / 1 + ---- 0 40 2/3 1/2 (1/3) 1/2 5 3 that happens to be equal to, -2/3 5 3 arctan(-----------) (2/3) 5 - 20 (1/3) (1/3) (1/3) (1/3) (2/3) + 2/3 5 ln(1 + 2 5 ) - 1/3 5 ln(1 - 2 5 + 4 5 ), alias, 0.9938377526645891192 Then c is irrational, and has irrationality measure at most, 1/2 4 ln(81 + 4 410 ) ---------------------------------, that equals, 23.144211575861624866 1/2 2 ln(81 + 4 410 ) - 6 - 3 ln(3) Proof: Consider 1 / | (3 n) 3 n | x (-x + 1) E(n) = | ----------------- dx | / 3 \(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) 3 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 (6 n - 5) (729 n - 1215 n + 325) E(n - 1) E(n) = 40/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1600 (3 n - 1) (3 n - 5) (n - 1) E(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n Subject to the initial conditions 6440 c E(0) = c, E(1) = ------ - 6400/3 3 and in Maple format E(n) = 40/3*(6*n-5)*(729*n^2-1215*n+325)/(3*n-4)/(3*n-2)/n*E(n-1)-1600*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*E(n-2) `Subject to the initial conditions` E(0) = c, E(1) = 6440/3*c-6400/3 It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence 2 (6 n - 5) (729 n - 1215 n + 325) B(n - 1) B(n) = 40/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1600 (3 n - 1) (3 n - 5) (n - 1) B(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions B(0) = 1, B(1) = 6440/3 and in Maple format It follows that the sequence of rational numbers, A(n), B(n), satisfy the same recurrence B(n) = 40/3*(6*n-5)*(729*n^2-1215*n+325)/(3*n-4)/(3*n-2)/n*B(n-1)-1600*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*B(n-2) `subject to the intial conditions` B(0) = 1, B(1) = 6440/3 and 2 (6 n - 5) (729 n - 1215 n + 325) A(n - 1) A(n) = 40/3 ------------------------------------------ (3 n - 4) (3 n - 2) n 1600 (3 n - 1) (3 n - 5) (n - 1) A(n - 2) - ----------------------------------------- (3 n - 4) (3 n - 2) n subject to the intial conditions A(0) = 0, A(1) = -6400/3 A(n) The sequence of rational numbers, ----, are good numerical approximations for c B(n) In Maple format: A(n) = 40/3*(6*n-5)*(729*n^2-1215*n+325)/(3*n-4)/(3*n-2)/n*A(n-1)-1600*(3*n-1)* (3*n-5)*(n-1)/(3*n-4)/(3*n-2)/n*A(n-2) `subject to the intial conditions` A(0) = 0, A(1) = -6400/3 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 1020251312250314887228037844484274307250079239767004585964367889806026646191\ 342345231791961165572614519870539950845059628521516433703925344099661022\ 092452753537551246599012560673140063797053211000341096803276366137200476\ 863226290650782706784248607143536063097955781071752542237180540297037362\ 851079674310977631622111541299196582627537907353843883066364255589767453\ 283580947721327618363655894589657931658460635877358587169992932632901969\ 569457702729396433159718945304168391084307707642625365555865783944888890\ 689608035509464740445860211461416703828866833943251626118394414125373477\ 463448480722169574159093418645854700229654058464636768828708963323658843\ 294232948546143404655891638821079771880110045508524083132252753379828987\ / 625778740605642475236877041068323278804375 / 1026577335701836731903405\ / 219103778673863315742262840709204832740806590654495342808345829857603558\ 521475215964704337761492906458598155365411685475826172176671172777793449\ 680559924354284241812382276774464888673968588771254334159850945655469266\ 659270549194707545208487141453131726337305499177816668424725840486619963\ 000752905728156644310872770649578211423049735393695172411656076054169678\ 313390898574048644777704131326023618528754742178229567553294376746936053\ 027646869957099243029027362181715912373717611198928170412845024609432520\ 964879519990106376011763199930847197380832587243531387223535714941525734\ 927602720365716948610014887037819112702099598465085301043228749424696234\ 523710082716858235663242875080959138621802347652447877915941709295026511\ 084416584032573245932 and its differene from c is 0.20851566957248598495198875189461940660076397370907998053939895301878321435\ 095957173442182188229850730957808879303905349084945017469722585739672342\ 967440155988959542476980818349576051460389967422981231124652680117493893\ 827708416369129383667537650861768470246914344848538707385414564522222476\ 242387309305973192235787899405787372169929163494233102506282630211515508\ 083997385776321642356726896619495427890715128783116855344439755819583015\ 596430011559649786752597259863927194114093756909924712888478533632849045\ 758744481647200517257271674619989165027960252128823817206685496087103378\ 913906519493192572861859134609422631310221495895601090076764080075343303\ 145493793082522371593622190593307787763823642227249865416259137209934893\ 334903595433636849114968486630178899289139972209333420892705583451979228\ 625254632762505219451875087324423168563489430025244172490187733250078125\ 783733640663449359928451815113923379833565013270968286544703002870661136\ 440692699192737861894403840302423992674916198587838019574614260933965014\ 633040851078743920905276008422257303793791820041611090394503606215094050\ 612047547961210746537403506301804962403661501802424125107135367336960605\ 007591233783665932980815996105679590030435038611035879430631540706792672\ 607491370344143258417695234585972737937364525670818356334889505932541191\ 833413686823398528396182119225865867093458155091091259117132338565598624\ 787967670084068161834773483859592357314545845419899312340652323814599957\ 103425017901110843771687155776301463441367316140570714336445519429748483\ 595622666327693230181707578137439839715439342977353008464987490148421091\ 925223075249984507832197844029437052822779779394327532504004779779356642\ 322177541059332143240746762226448155724512545985932172924842604358806545\ 247681638876467152942130669472538011987055142500880213956794048044065620\ 315829564935646137537856687871995701893540366980469684831385899467173783\ 312181729954100120446243790778882662493769832798670318548231520032865445\ 547835653169156806351647629822783288170520953187356497435696388295734534\ 605195535703492207840626447829004211486743874746214936296855236095978400\ 526574808734153106169838730146307794171431741685551872880478122049814075\ 217700526801112944151854067396034133968563951668496040434727686338040223\ 213880371228614762869319705575548325154810255250678596886875867775744674\ 400977728750408444605817968219734935780579050500595833090268745561924311\ 980268143412853924799412252358397423329690943342905956017385012186535372\ 527555566748522451927184071219672681630722923621602734475959601268370535\ 099681845377345178256099892066624747633448867534103743242484461027708324\ 749994718733950479915833678653061585636724077984519115003128997007606447\ 403349407525684665795326363481984027007560298777549967373887114462990425\ 345356670721250973531067790023884320779375244867024732097834203661117266\ 065904641724081825787520752564730025185010421841910825891532773708972455\ 119736912102501457754220225234116377315115670815107883060613584355398292\ 940580662537830647365996546531501266110710532444244224056588180800651346\ 186796709190004624809340831191618570872097571215415805621939814317019934\ 812397073180127253355293661465566401986908819300126506185250976230299395\ 623505149681648113455969016883046953803788012592772614417141203928762362\ 874981665849495322800928417549455086916424731609890546611670924462784235\ 676859430551809015544865805928688977741962877307390934870842156389921166\ 544339683238671377600298681405940053560342026029267524707629154116225545\ 516685698661743637409460831836783161939665446257542700064934641573127460\ 365040715690340759558134374290823141209646813763240406337494146593924581\ 323310875091711649476796256751378075100399497932862261669635792665127309\ 015525419023295236054784408481583571551746772195857107145114832999453269\ 412411949151323783312249494773122420917059075953666732951980830084899600\ 764797777835659522073405678666227399893520185785888838469494418300906215\ 375196778149846762460495803512654719813937897431937727926981151909721408\ 491910644676665552463399628115191401304046867122559628819361723301990873\ 109528766530332477187304242745700369594662959590707599726622483657260222\ 786091663179701669478093682657434781501349675429537078792301592488772525\ 620486016523998339014539129745454317919057935258777834704790362823306662\ 019675871446135122795349173148978862189546705429822608468479013967192112\ 587196706557112857894092825876896320053605926582971000917007242352809846\ 114429268987367570377018713328122993523573696461431474586394611789773144\ 095400817863824548342112111572492145648487849689267401185778524910233008\ 785814721932862451407405610905984053408858207029845134819336702739312545\ 182688266213296085269042725239198991566770013556236211291912850539620824\ 916329285692610135844481003052743743006674240826386130844945181296630968\ 459551978481294541128837137210663256892986568659916553362200314558698300\ 906672237362159293306833289350213012520371153068917380550389080603552516\ 019090301431185129757087841264385738444269031553443561422522810886749072\ 098193729209927145973759428808560446754884393300147383147419306832770115\ 649394457410457331129922868876696882008946144448076408902170867105775323\ 510457060079742195548564215118188358493546426935928174167996990666327173\ 095978888970949261224873182106471184929211016916707955520489702370844741\ 082942784383470667658689630804449096958897960269445180554677493770395480\ 360616525820939445957900218093493378419468195587217702238253756635155908\ 247142921148718335803754807804603354100081924971924980995552795509207767\ 436931145667602093200910206460900775294271185578800976440901768174896942\ 324488854436234014766460929993986672583017365969187620244889484321272864\ 338040054319149731714823168118315246544649277053416120404311177566893948\ 977530712496557265572729441000503747870404272675060928062551244795973046\ 797889347252208997529276627635712192616761593240267936237371297992660007\ 014409305701309730753474640897261743143168368793346958888611991021952549\ 985165435132024324319984883917691502872991918743547274662275599104719287\ 222780520935017239745586890707313257525823172980642866307672741939428189\ -883 1478097933442509277083975809605931807151630697225353984902184495645 10 The smallest empirical delta from, 100, to , 200, is 0.1439237232 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 3 n 3 n trunc(---) trunc(---) 4 4 A(n) d(3 n) 9 B(n) d(3 n) 9 Lemma: , -----------------------, and , -----------------------, n n 40 40 are always integers Let's call these new integer sequences, A1(n), B1(n) Using the above recurrence, by the Poincare lemma, the rate of growth of A(n\ ), B(n) are (essentially) 1/2 n C (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 (21 + 12 3 )| |---------------------| | 1/2 | \-65600 + 3240 410 / Hence | A1(n) | C | c - ----- | <= ------------------------ | B1(n) | / 1/2 1/2 \n |410 (21 + 12 3 )| |---------------------| | 1/2 | \-65600 + 3240 410 / 3 n trunc(---) 4 B(n) d(3 n) 9 But , B1(n) = -----------------------, hence n 40 1/2 (1/4) n B1(n), is of the order , (40/9 exp(3) (21 + 12 3 ) 9 ) 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 1/4 1/2 40 9 ln(3240 + 160 410 ) + 3 - ln(-------) 9 That in floating-point is, 0.045158517 It follows that an irrationality measure for c is 1/2 4 ln(81 + 4 410 ) --------------------------------- 1/2 2 ln(81 + 4 410 ) - 6 - 3 ln(3) that equals, approximately 23.14421700 ------------------------------------------ ----------------------------------------------- This ends this book that took, 1359.468, seconds to generate.