# 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
2 years, 2 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.

- 2 years, 2 months ago

what is the inverse of grahams number

- 1 year, 1 month ago

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

- 1 year, 1 month ago

Testing Math Editor: $\pi$

- 2 years, 1 month ago

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

- 2 years, 1 month ago

what's the algorithm?

- 2 years, 2 months ago

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.

- 2 years, 2 months ago

3.345625467385246375823675867348259426395436858685673245678845673333302367489236758493678624396724839674839678492456789106574385627384562385962785967289674280654803333221...

- 1 year, 1 month ago

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

- 1 year, 1 month ago

started decimal place 46374

- 1 year, 1 month ago

it had a lot of odds and 1 even

- 1 year, 1 month ago

another: 4.12344678259467589436578249365782493333331029564782936758293672892345636758935674392567238962789434247148245623392564839578123456123456925379267896661154673845268012345679213....

- 1 year, 1 month ago

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

- 1 year, 1 month ago

started decimal place 142857

- 1 year, 1 month ago

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)

- 1 year, 1 month ago

period triples

- 1 year, 1 month ago

546738111123/999999999999

- 1 year, 1 month ago

23456/99999 (period=5)

- 1 year, 1 month ago

5.15673842536784536758786946875047894037580204308200206392579239563853633333026790678042606596503768034768013999761111124910123...

- 1 year, 1 month ago

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

- 1 year, 1 month ago

started decimal place 999001

- 1 year, 1 month ago

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

- 1 year, 1 month ago

It's divisible by 7

- 1 year, 1 month ago

6.546728567584673256743295627438956784396578234967329647812967489367850163478046780654738563478056237480628111333999777546738146713294672946170856173805617438056173805637850613785061795714380561379461378463785016437586071483674820365780476892071520773457215893657896574839678964723911011...

- 1 year, 1 month ago

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

- 1 year, 1 month ago

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)

- 1 year, 1 month ago

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

- 1 year, 1 month ago

what a periodic sequence

- 1 year, 1 month ago

5.6574891673895617834523784016478301357835631756347827468203567813647813257803657830573205417329536780513728140613751376037805617834178203567318065354721054273805723104617357328033336217304617358016785203675830682306473280567281036478036121111615783236718036701561111116163781526783106732806512051627780254167394394567956293567239456379452367945231755547956371453925781936478329615782306712065720567203...

- 1 year, 1 month ago

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

- 1 year ago

hi

- 2 years, 2 months ago