# Last 10000 digits of Graham's number

So, if you didn't know, it's possible to compute the last $$n$$ digits of the Graham's number quite easily, by observing the fact that

$3\uparrow3=...7$,

$3\uparrow3\uparrow3=...87$,

$3\uparrow3\uparrow3\uparrow3=...387$,

$3\uparrow3\uparrow3\uparrow3\uparrow3=...5387$, and so on.

In fact, if you search on the internet you'll find several people that computed up to $200$ digits, $400$ digits, or $500$ digits.

Now, I admit I was pretty disappointed with this results. I couldn't find anyone that pushed up the calculations! (I may be wrong, in that case show me if anyone else actually calculated more than that)

So, I developed an algorithm that is sufficiently efficient to compute the first $500$ digits in only approximately $2$ seconds. With this algorithm, I was able to compute the last $10000$ digits of Graham's number on my computer, in $16316.5$ seconds (approximately $2.2$ hours).

I'm not entirely sure how, but I'm pretty sure it's possible to make the algorithm much more efficient than that, so that even calculating these digits should happen in a matter of few seconds. If you have any good ideas, tell me below (try it first, if possible).

Either way, here they are (new line every 70 digits):

$g_{64} = ...3078726077030301309631860565499591166728394144521822940211684925744690\\ 8303160862621307856182261004203129683467872420025335707165042882886038\\ 1527890583177734743488999363221709377188705302161340082623104165260987\\ 2469522381180520893701095105746695201447806881045566940953003050207963\\ 0251089314581777820684968377573229456578469591751099452615715258153133\\ 8317144176357246277980971517349406785579279353063617993752257361282030\\ 1473864489406090828511196812234883812826536412935235758505566522273299\\ 8513867089855758447648371115779454007188631486534611854130768464954083\\ 8335805841695412280766038020711553526827091695879475106425076890327782\\ 6708848390877435531688133831988779505683625673270427786212688069705881\\ 0276174028639378952134276596828174174610570754797760763975177038244691\\ 2063024310915173151554672020720329805779214569991795690518659602446902\\ 7274217279141430415867319657287140268008652315291316261820652195021921\\ 0914610704519073926283967434339662068326819744974641935341502976180597\\ 5219746989916790553951240749624022306775365113200227817050284227367149\\ 7491794032802699070161003317178855043208655046184676579497958334888729\\ 3809617659827235067373513656241299335615924204033665860263764635136445\\ 0901965169912468031701035813068048871232519853582991620638263170147783\\ 2698324585503287762867838791720029901423547073086007824609234282758224\\ 9084362130009203937656462657958086964494023780323916935845145868457435\\ 0019308749993296237589317856219609033942624384808517627658437282947072\\ 8254599948415970659821958864982353541909598033354078979538256357245359\\ 7473687377205449098623941105789048433603977408157821289037966848053431\\ 0161245447459165410165069396137727188277544585197396781572979626659473\\ 8946097169571922151724219229759670392571616731391424780277194256133384\\ 1909070301383939429023145632838588124976137279502774086096179244679618\\ 6025376358215497336320381116685826498270685734807990885455869123699181\\ 9878323695993552951130721247635787520841764173839071101104122258672751\\ 9228928871964307453282154261444871750080952421236363768139610777495235\\ 3201683350142428109136897852904313178813533335511088902235947721071584\\ 9496037260619984908856098462387682166967457951388588832356929998573679\\ 5420111111854634404435045676794013567654575959588995123046187234360352\\ 1793889196205026750886423489731195416265541685656346013519486952120940\\ 3751415314099407271948277374441254695053697070411390864858720859625554\\ 1959030238060629137169138722855477794398025076773239670591720768747467\\ 2844654063418138034129430436097660215542336711205739081742346117574411\\ 8240134002338774237104408010940301648706232749183781478765763754598101\\ 5382140877149223264515258326901107991667754434423538374815474031008806\\ 3794659755273784075854629840335173116218238312437640446823996561592082\\ 8447876334410730503146643034564958892386400206720157774336643971046097\\ 1913397786711927076109359761269701118640088324214111573518542523175301\\ 8437333602444678510859228109479432808176387274946992467944754462151350\\ 5367844082852088230997958080227897832298874619826542016735413058821008\\ 1847428327743235541621704680913967417772759547636593847461261352062528\\ 6493615579933087865883727503023169846062624381686315823768546266806694\\ 4052359344981816230189301309354006700248433587219900315863936527371581\\ 9660118267438193855827192236488406202498847093959194410120314045227731\\ 5157935152536936493425053093270987737803549232774078167979402086740138\\ 2441063153368792024085513096521839198783970159526194524749783176345096\\ 6138354215340807779927410436230549205899966191897479395002808032068505\\ 2798877144984365085442674905448699279843309239159997374495902995758836\\ 3427364161563026748153852589303542350028921428192299937990553925840346\\ 3179740764550809001835012576720025421347754368577841258214793482088528\\ 3528473446395144549774006915475700147362629356396411599745063763259120\\ 7602085065331518191297627524916528283327470445487220431209794181800837\\ 5242881807867409536257070239161286255743400994204946286838833646124751\\ 9450140740102350858410502955539960188235779958081190441955895515413996\\ 2907362641416032599108674402147606737302706859009869989241966114092477\\ 2407789976751585993614061850356986544196252382034927589676238991215557\\ 8594439247420428277018540432967913180650401616546821719644550936450059\\ 5583090752015617577969861470976788938740200600298288287888330891308863\\ 3722819728625451800838400216724842196639874673295077669833180844718217\\ 9341171005320299673923404289980856827543457807278033622507455463530345\\ 2502047026773318085940276837031308524437227254648631365299090852540852\\ 7368117005989256614980159639267250074517606853740135934062100746075654\\ 0813896593247352620824016619700186551382187297721949846316900749082737\\ 8094264056082839271636552848839603826288292359895120699259424392975074\\ 9821816437833463246454551763732757655276760590582693972496924532998092\\ 3380190551831802463416887618541048817756223855531234022957969144122483\\ 6406466849925916755964304671002296817244339224792182135995868969341492\\ 3690776243056770164822502041983470306716296896922618604046908954485889\\ 1348275274667315520855871416643581761678324893080737347771500148013131\\ 5872093173217027881014846523723198056977435033203823397583204572679056\\ 6651633842068281545694928486633422445461995837450720456764508778883440\\ 3854173232861131892325939242549724454004862230630447361306160309037871\\ 5800584480793579521136958705250576661572845777934961160672813467687918\\ 2468219518101142821666186785543970961610009398295493882433280467468806\\ 9413764541931265604057917863711717794304584231021596233589764804054811\\ 8226228812074485140806951381912453398593093071760684032101862850540079\\ 7648371363380891593720550471092872187099316006161890897978303030444170\\ 7453650131619894698610993458668339554364084020023399067051642966934860\\ 0027570319308957395061221345848777985085039705997484485916440720079353\\ 6749317556015986374673052074784015605376345910038695996792166640248506\\ 5151972178292734011493177498161942761055395341214573854386960142347615\\ 1064120816786711566627026981412394927353467365863136279136411085250120\\ 5029022957845553710087982096902106709311969126401168909948606316616230\\ 2923902195299675526555986392068330814711696071435279166352601845751091\\ 1413391101513558310051637274156744456218580403883246348766716791121059\\ 6640361544098108142733814394129952702989282677057681983318348451157632\\ 3445763461588558180723948837731724171820914217827456505430234904693165\\ 7016237345668584450386474453301178678720075637289970867637583657220506\\ 2491301610433455913955841065225929205594669325571087621309871185323210\\ 5196699704725638064513028255040447297723752850180331481056404968157353\\ 9061785278598760313140865068530080919200003079350549373708348792972286\\ 8670573755820995979960622710931298072888031118784826564331478394544516\\ 8149803763580948273619215580715203535826455618889996767810934669730243\\ 5416969251647217062623523969036088524240889973797522976568651221236460\\ 6700882735550252805162688717743002242430178153096139879227557097022144\\ 3057790329335567051650934703248833671211260215491243781456055632180252\\ 2593943710111389180996687955300606304861015768885793044484950849027511\\ 0106029803619475963780866084863862122324538632723342196159851036070658\\ 3695251304412583544807924551217328349121977645145493850412041271265386\\ 4611628171304512992288612473168928694708607916093810579534245595047488\\ 6617927058056116386911212141737555332931772421804354502689741745136461\\ 9149258184608873236073683901075275381240402732984629660210771321778593\\ 5317908316720145478172167767818370069254396358778477381821832282338662\\ 9557597665763526037469623882734970261718460508041643967446394822226604\\ 7262800922130068469037144793383117028262382841196033782589178075561744\\ 9686306276313945559616369461957875659406855496062664521105020034463607\\ 8639152176612841273246755062872676148243079798928244275312027774818688\\ 7250109520756590181937968438911972868200142926836525620315535316031054\\ 6891685301933822474738169794305170907203165619769535061113213915856256\\ 1824661400448048835400453257011706951826263434793588413474586938545913\\ 1913595602946034912375719564262285371586081255164508962613460090798485\\ 7847797205307145186514675412317888384738009343344165376733605639874152\\ 6838837135702194865074959666167436192933645884998056100697104793100679\\ 4152084453613830911021630017437654919684883920437258419601503784784516\\ 0671512017198801157547084883939593053650556078872159994750221442214834\\ 8268144787270731001365537383577746098505866012640076129423352326255313\\ 3073942052007839547749762554111899859772880815945865752809988634672233\\ 4769804780146302789353612329312586963866559329949214911489134763214665\\ 4314303272656947761889503867538372033508034358690038674211367316517236\\ 2113256247997506770294235705056911305065974352655256553654276889526636\\ 0391135992668989734244822601493574507744556050638326609473542254360350\\ 8674855342484610627305685534794791282019520577643564769466316663822950\\ 0048280051827615363513800094323248679021061702425944029209484941954536\\ 7418064519308105163357496871638118822504114501587037019405680648005022\\ 5768533805530305183368091271811490817539484300268084104379556148104831\\ 5835447210850384076723823375354333111031697890169996590703687564769571\\ 4199517294684058268271081207938885760678089057660597351282040660918730\\ 7108483992113117957918089160673029776868734932638038255189701221105348\\ 1886141584874851920098526106525203948232207371149341083916873785440379\\ 8603368448472052729248390757866617805529414157119366603081892881936678\\ 7741482317801728126934985735783270950758576591974947039193152967596669\\ 2340488030236244704910353178090822611674695077464191287728244330583239\\ 5092525499355092526168572459565741317934416750148502425950695064738395\\ 6574791365193517983345353625214300354012602677162267216041981065226316\\ 9355188780388144831406525261687850955526460510711720009970929124954437\\ 8887496062882911725063001303622934916080254594614945788714278323508292\\ 4210209182589675356043086993801689249889268099510169055919951195027887\\ 1783083701834023647454888222216157322801013297450927344594504343300901\\ 0969280253527518332898844615089404248265018193851562535796399618993967\\ 905496638003222348723967018485186439059104575627262464195387$

Note by Aldo Roberto Pessolano
1 year, 6 months ago

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## Comments

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Top Newest

1.128283173717379590959517773101717618496378493678149367192657839641789463786437859367489657329467381596378406378057381940785901674023748023671802367148023647820367328407895036274810738406738407384063578032748036258407389506328051020408160904010025062523549823647839567194678329678162835063810463278047328140333316703267023647382067124036721507132940738947389432691034758903751932075819403673027810635695679579693103446748617840672046738056713046783210467328036748105780637840632758023456167498146273849326758946738239215623749123401... is irrational approximations 9/8,35/31

- 3 months, 4 weeks ago

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5.6574891673895617834523784016478301357835631756347827468203567813647813257803657830573205417329536780513728140613751376037805617834178203567318065354721054273805723104617357328033336217304617358016785203675830682306473280567281036478036121111615783236718036701561111116163781526783106732806512051627780254167394394567956293567239456379452367945231755547956371453925781936478329615782306712065720567203...

- 4 months, 1 week ago

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3,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,... (period=2) but i have a 3 at the start

- 4 months, 2 weeks ago

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what a periodic sequence

- 4 months, 2 weeks ago

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periodic sequence: 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,... (period=2)

- 4 months, 2 weeks ago

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6.546728567584673256743295627438956784396578234967329647812967489367850163478046780654738563478056237480628111333999777546738146713294672946170856173805617438056173805637850613785061795714380561379461378463785016437586071483674820365780476892071520773457215893657896574839678964723911011...

- 4 months, 2 weeks ago

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you could see the digits: ......658963725933333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333674382956743286972678594310001647193......

- 4 months, 2 weeks ago

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umm the number 26378146328956830967280674283967389657849306784230673289567807685240677773333506342657802367819567483578695487530247483120657483075101 is this prime? try in Number world

- 4 months, 2 weeks ago

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It's divisible by 7

- 4 months, 2 weeks ago

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5.15673842536784536758786946875047894037580204308200206392579239563853633333026790678042606596503768034768013999761111124910123...

- 4 months, 2 weeks ago

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find the digits ......3159735193759135793133331759173979973197351973113373313579197315793139735193791735931597391375975973797735179...... (all the digits here were odd)

- 4 months, 2 weeks ago

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started decimal place 999001

- 4 months, 2 weeks ago

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23456/99999 (period=5)

- 4 months, 2 weeks ago

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546738111123/999999999999

- 4 months, 2 weeks ago

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1/3=0.33333333333333333333333333... (period 1),1/9=0.111111111111111111111111111111... (period also is 1),1/27=0.037037037037037037037037037037037037... (period=3),1/81=0.01234567901234567901234567901234567901234567901... (period=9),...,1/grahams number (period=grahams number/9)

- 4 months, 2 weeks ago

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period triples

- 4 months, 2 weeks ago

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another: 4.12344678259467589436578249365782493333331029564782936758293672892345636758935674392567238962789434247148245623392564839578123456123456925379267896661154673845268012345679213....

- 4 months, 2 weeks ago

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look at the digits .....56278524678333333333333...(100 3s total)...333333335326845367823524832352784325682283683678335683.....

- 4 months, 2 weeks ago

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started decimal place 142857

- 4 months, 2 weeks ago

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3.345625467385246375823675867348259426395436858685673245678845673333302367489236758493678624396724839674839678492456789106574385627384562385962785967289674280654803333221...

- 4 months, 2 weeks ago

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what a constant look at the digits ......31579315973175917973999717329......

- 4 months, 2 weeks ago

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it had a lot of odds and 1 even

- 4 months, 2 weeks ago

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started decimal place 46374

- 4 months, 2 weeks ago

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Testing Math Editor: $\pi$

- 1 year, 4 months ago

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Booya! $e^{i \pi} +1 = 0$

- 1 year, 4 months ago

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what's the algorithm?

- 1 year, 5 months ago

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The most efficient algorithm I thought for now is just one line on Mathematica:

calc = 3; Do[calc = PowerMod[3, calc, 10^i], {i, 1, 500}]; calc

This computes the latest 500 digits correctly in roughly 0.7 seconds on my computer. For 1000 digits, it takes 6.2 seconds. The main problem with this algorithm is that it gets progressively much slower since it computes all the already computed digits with every calculation. There must be a way to avoid recomputing all the digits every time, for example to get the 1001st digit pretty much instantly just by knowing the latest 1000 digits, but I can't quite figure out how.

- 1 year, 5 months ago

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hi

- 1 year, 5 months ago

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Update: I just found a site that lists exactly the same digits, so I'm definitely not the first to have calculated this digits. Still, the efficiency question remains.

- 1 year, 6 months ago

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1/243=0.004115226337448559670781893004115226337448559670781893004.... , period=27, and etc

- 4 months, 2 weeks ago

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what is the inverse of grahams number

- 4 months, 2 weeks ago

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