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
3 weeks, 1 day 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.

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hi

Irina Richards - 1 week, 1 day ago

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what's the algorithm?

Jaydev Singh - 6 days, 22 hours 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.

Aldo Roberto Pessolano - 6 days, 21 hours ago

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