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

33=...73\uparrow3=...7,

333=...873\uparrow3\uparrow3=...87,

3333=...3873\uparrow3\uparrow3\uparrow3=...387,

33333=...53873\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 200200 digits, 400400 digits, or 500500 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 500500 digits in only approximately 22 seconds. With this algorithm, I was able to compute the last 1000010000 digits of Graham's number on my computer, in 16316.516316.5 seconds (approximately 2.22.2 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):

g64=...3078726077030301309631860565499591166728394144521822940211684925744690830316086262130785618226100420312968346787242002533570716504288288603815278905831777347434889993632217093771887053021613400826231041652609872469522381180520893701095105746695201447806881045566940953003050207963025108931458177782068496837757322945657846959175109945261571525815313383171441763572462779809715173494067855792793530636179937522573612820301473864489406090828511196812234883812826536412935235758505566522273299851386708985575844764837111577945400718863148653461185413076846495408383358058416954122807660380207115535268270916958794751064250768903277826708848390877435531688133831988779505683625673270427786212688069705881027617402863937895213427659682817417461057075479776076397517703824469120630243109151731515546720207203298057792145699917956905186596024469027274217279141430415867319657287140268008652315291316261820652195021921091461070451907392628396743433966206832681974497464193534150297618059752197469899167905539512407496240223067753651132002278170502842273671497491794032802699070161003317178855043208655046184676579497958334888729380961765982723506737351365624129933561592420403366586026376463513644509019651699124680317010358130680488712325198535829916206382631701477832698324585503287762867838791720029901423547073086007824609234282758224908436213000920393765646265795808696449402378032391693584514586845743500193087499932962375893178562196090339426243848085176276584372829470728254599948415970659821958864982353541909598033354078979538256357245359747368737720544909862394110578904843360397740815782128903796684805343101612454474591654101650693961377271882775445851973967815729796266594738946097169571922151724219229759670392571616731391424780277194256133384190907030138393942902314563283858812497613727950277408609617924467961860253763582154973363203811166858264982706857348079908854558691236991819878323695993552951130721247635787520841764173839071101104122258672751922892887196430745328215426144487175008095242123636376813961077749523532016833501424281091368978529043131788135333355110889022359477210715849496037260619984908856098462387682166967457951388588832356929998573679542011111185463440443504567679401356765457595958899512304618723436035217938891962050267508864234897311954162655416856563460135194869521209403751415314099407271948277374441254695053697070411390864858720859625554195903023806062913716913872285547779439802507677323967059172076874746728446540634181380341294304360976602155423367112057390817423461175744118240134002338774237104408010940301648706232749183781478765763754598101538214087714922326451525832690110799166775443442353837481547403100880637946597552737840758546298403351731162182383124376404468239965615920828447876334410730503146643034564958892386400206720157774336643971046097191339778671192707610935976126970111864008832421411157351854252317530184373336024446785108592281094794328081763872749469924679447544621513505367844082852088230997958080227897832298874619826542016735413058821008184742832774323554162170468091396741777275954763659384746126135206252864936155799330878658837275030231698460626243816863158237685462668066944052359344981816230189301309354006700248433587219900315863936527371581966011826743819385582719223648840620249884709395919441012031404522773151579351525369364934250530932709877378035492327740781679794020867401382441063153368792024085513096521839198783970159526194524749783176345096613835421534080777992741043623054920589996619189747939500280803206850527988771449843650854426749054486992798433092391599973744959029957588363427364161563026748153852589303542350028921428192299937990553925840346317974076455080900183501257672002542134775436857784125821479348208852835284734463951445497740069154757001473626293563964115997450637632591207602085065331518191297627524916528283327470445487220431209794181800837524288180786740953625707023916128625574340099420494628683883364612475194501407401023508584105029555399601882357799580811904419558955154139962907362641416032599108674402147606737302706859009869989241966114092477240778997675158599361406185035698654419625238203492758967623899121555785944392474204282770185404329679131806504016165468217196445509364500595583090752015617577969861470976788938740200600298288287888330891308863372281972862545180083840021672484219663987467329507766983318084471821793411710053202996739234042899808568275434578072780336225074554635303452502047026773318085940276837031308524437227254648631365299090852540852736811700598925661498015963926725007451760685374013593406210074607565408138965932473526208240166197001865513821872977219498463169007490827378094264056082839271636552848839603826288292359895120699259424392975074982181643783346324645455176373275765527676059058269397249692453299809233801905518318024634168876185410488177562238555312340229579691441224836406466849925916755964304671002296817244339224792182135995868969341492369077624305677016482250204198347030671629689692261860404690895448588913482752746673155208558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Note by Aldo Roberto Pessolano
1 year, 4 months ago

No vote yet
1 vote

  Easy Math Editor

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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.

Aldo Roberto Pessolano - 1 year, 4 months ago

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what is the inverse of grahams number

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week ago

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Testing Math Editor: π \pi

Brett Washbrook - 1 year, 3 months ago

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Booya! eiπ+1=0e^{i \pi} +1 = 0

Brett Washbrook - 1 year, 3 months ago

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what's the algorithm?

Jaydev Singh - 1 year, 4 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.

Aldo Roberto Pessolano - 1 year, 4 months ago

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3.345625467385246375823675867348259426395436858685673245678845673333302367489236758493678624396724839674839678492456789106574385627384562385962785967289674280654803333221...

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week ago

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started decimal place 46374

arjon arapi - 3 months, 1 week ago

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it had a lot of odds and 1 even

arjon arapi - 3 months, 1 week ago

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another: 4.12344678259467589436578249365782493333331029564782936758293672892345636758935674392567238962789434247148245623392564839578123456123456925379267896661154673845268012345679213....

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week ago

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started decimal place 142857

arjon arapi - 3 months, 1 week 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)

arjon arapi - 3 months, 1 week ago

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period triples

arjon arapi - 3 months, 1 week ago

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546738111123/999999999999

arjon arapi - 3 months, 1 week ago

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23456/99999 (period=5)

arjon arapi - 3 months, 1 week ago

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5.15673842536784536758786946875047894037580204308200206392579239563853633333026790678042606596503768034768013999761111124910123...

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week ago

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started decimal place 999001

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week ago

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It's divisible by 7

Aldo Roberto Pessolano - 3 months, 1 week ago

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6.546728567584673256743295627438956784396578234967329647812967489367850163478046780654738563478056237480628111333999777546738146713294672946170856173805617438056173805637850613785061795714380561379461378463785016437586071483674820365780476892071520773457215893657896574839678964723911011...

arjon arapi - 3 months, 1 week ago

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

arjon arapi - 3 months, 1 week 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)

arjon arapi - 3 months 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

arjon arapi - 3 months ago

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what a periodic sequence

arjon arapi - 3 months ago

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5.6574891673895617834523784016478301357835631756347827468203567813647813257803657830573205417329536780513728140613751376037805617834178203567318065354721054273805723104617357328033336217304617358016785203675830682306473280567281036478036121111615783236718036701561111116163781526783106732806512051627780254167394394567956293567239456379452367945231755547956371453925781936478329615782306712065720567203...

arjon arapi - 3 months ago

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

arjon arapi - 2 months, 2 weeks ago

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hi

Irina Richards - 1 year, 4 months ago

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