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$

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Last 10000 digits of Graham's number

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