1010,000,000,000
10^^3, a trialogue.
10100,000,000,000
370110361............6611
3.70110361... x 10297,121,486,764
11^^3.
1.31137265... x 10847,288,609,430
(10^847288609443 - 1)/7625597484987. Can be expressed as r(3, 27) using the reptend function. Its decimal expansion is: 13113726523970925.........159918811111111111124224837635082036.........271029922222222222235335948746193147.........382141033333333333346447059857304258.........49325214444444444445755817096841.........60436325555555555556866928207952.........71547436666666666667978039319063.........82658547777777777779089150430174......0488077
125801429062749131786039069820121551............905650177570718344711077047886315075206738945776100739387
~ 1.2580142906... x 103,638,334,640,024
3^(3^(3^3)). Can also be expressed as 37,625,597,484,987, 33^27, or 3^^4.
4969981598............016
4.969981598... x 109,622,088,391,634
12^^3.
1010,000,000,000,000
10100,000,000,000,000
1.2322... * 10264,007,110,309,345
1.5^^14
A power tower of 14 1.5s.
101,000,000,000,000,000
5.95311951612318355925717768203975499... x 10208,309,294,085,508,228
5^5^5^2.
1.322449... x 1022,212,093,154,093,428,529
(2^^3)^^3. Can also be expressed as 22^66.
279,641,170,620,168,673,833 ≈ 5.076025219... x 1023,974,381,246,463,762,439
This power of 2 has the same first 19 digits (and the same number of digits) as 350,247,984,153,525,417,450, the next entry. It starts with 5076025219062843177372568120346456..., and ends with ...8587147886872589316067950592. Below are more digits of this number.
5076025219062843177372568120346456935927492816254654790984064653519938602128051975679885879975580541765574009637650907195596863497405792871315669251518673955865859512297940530749576085592457044498319667847360873771617172575569546181956948620082218406839134172134693415904915620058635829457233050366816215795145160356036600669738183752476667835026594201793412520565721382905112556892172505685126763004615266903861759421465538732484357686815457863036097023385824203326083136963426241031811……………………………………………………………………………………………………………………………………………………………………………………... 45075036760278939992059369280492581033539282763085950607445348716061055760775692009947406394696219320249399727251506224600004884515946140322274051574741421945480649779256858587147886872589316067950592
In this and the next entry, the few digits after the identical leading digits are in red to highlight the difference.
350,247,984,153,525,417,450 ≈ 5.076025219... x 1023,974,381,246,463,762,439
This power of 3 is almost exactly equal to 279,641,170,620,168,673,833. It starts with 5076025219062843177406..., and ends with ...9374942249. It is approximately 1.00000000000000000000672 times 279,641,170,620,168.673,833. Below are more digits of this number.
5076025219062843177406678353203677516510936923793095250465635995537448249966593136547270101630625423591430098651531611516103466445594026633219245202613498464614072402169354219580023311818978334013780534100551003014183407203012909503849649838923286250988840786548498574052336990423486584341459428948834186492934637453987073004396864203641527965474985189240010730722635747810694534949057140074651216553506718140181098034508072067759722909472555062941561978392340682747108103736383786459278……………………………………………………………………………………………………………………………………………………………………………………….9374942249
2.285367... x 10381,600,854,690,147,056,244,358,827,360
2^2^100, the googolplexibit. The last 10 digits of this number are ...3335387136.
5454312............7493833727
5.454312... x 1051,217,599,719,369,681,875,006,054,625,051,616,349
The fifth term in the Catalan-Mersenne sequence. Being greater than 1010^37, this number is far too large to compute its complete decimal expansion, let alone test if it's prime using currently known methods.
1.1899811685... x 10102,435,199,438,739,363,750,012,109,250,103,232,700
2^2^2^7. It starts with 118998116852966565343067... and ends with ...53903192091641511936.
The Googolplex Range (10^10^80 ~ 10^10^500)
2.5517890642... x 103010299956639811952137388947244930267681898814621085413104274611271081892744245094869272521181861720
The googolth power of 2. It begins with 2551789064200187957632800640620918466171..., and ends with ...3890995893380022607743740081787109376. I actually have 74,000 digits of this number on a wall somewhere in my house.
Back when I first got interested in large numbers, I used to do all my calculations on the Windows calculator. I erroneously obtained 1226771622962442794288155796038821007259530432563079... as the leading digits of this number when I first tried the calculation, but in early May of 2014, the result somehow changed to 3655853..., and I discovered that the Windows calculator was only accurate to about 38 digits (128 bits) with logarithms. The calculator app's accuracy has since improved to around 47 digits.
1.2724681704... x 104771212547196624372950279032551153092001288641906958648298656403052291527836611230429683556476163015
Googolth power of 3. The last digits are ...65522 followed by 100 zeros, and then a 1.
3.9188192081... x 106989700043360188047862611052755069732318101185378914586895725388728918107255754905130727478818138279
Googolth power of 5. Multiplying this number by the entry before the last results in a googolplex.
1010^100
A googolplex. Can also be expressed as 10^10^10^2. Along with the googol, it is one of the two quintessential large numbers. The most common description of the googolplex's size is that it would not be possible to write out all the zeros even if a digit could be written on a single atom.
Some factors of numbers just above googolplex have been found, such as:
googolplex + 1 = 316912650057057350374175801344000001 x 155409907106060194289411023528840396801 x 1467607904432329964944690923937202176001 x 11438565962996913740067907829760000000001 x 495176015714152109959649689600000000000001 x 7399415202816574979127045311692800000000001 x 9823157208340761024963422324575436800000001 x 153011449181500580269468675253...004467201
googolplex + 2 = 2 x 3 x 166666666666666666666666666666...666666666666666666666666666667 (10100 digits in the final factor)
googolplex + 3 = 7 x 1429 x 999700089973008097570728781365...8916325102469259222233330001 (10100 - 4 digits in the final factor)
googolplex + 4 = 22 x 1129 x 1181 x 18917 x 54917 x 14950993 x 2936848501 x 63822551333 x ...
googolplex + 5 = 3 x 5 x 29 x 666666667 x 344827...2069
10^10^100 + 37 is the first integer after 10^10^100 with no known prime factors. It could plausibly be the first prime greater than googolplex, although we may never know for sure. In fact, for n = 37, 111, 133, 183, 187, 193, 331, 499, 511, and 519, no factors of 10^10^100 + n are known.
1.0016338457... x 1010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
320959032742893846042965675220214012506075180067979301169235453386341774775719406287167658023089812370, the first power of 3 that is greater than a googolplex. It is very close to a googolplex in terms of ratio (in fact it is less than 1.002 times a googolplex). It starts with 1001633845720065037245058736407..., and ends with ...5287979849.
1.0023777406... x 1010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
233,219,280,948,873,623,478,703,194,294,893,901,758,648,313,930,245,806,120,547,563,958,159,347,766,086,252,158,501,397,433,593,701,551. It is the first power of 2 to be greater than a googolplex. Coincidentally, it is very close to a googolplex (in fact even closer in terms of ratio than 1024 is to 1000) It starts with 100237774068258191291356652609799417599067..., and it ends with ...215599182339992927666157787971954826805248. More digits are given below.
100237774068258191291356652609799417599067343177599679075853090406145885491297965766192391921808372207511666330351479842825690694045923065369420716156150106516721534771683438379710931765831859029500096278333967134504248458798713324431058768246620528030281025712858245475129072285125516670526408042732689668962555802951707932337674892866805183099975191984453558986140352179091036611149246971005741711...215599182339992927666157787971954826805248
7.0842519670... x 1015684240429131529254685698284890751184639406145730291592802676915731672495230992603635422093849214947
The reptend in the reciprocal of 3^212. Can be expressed as r(3, 212) using the reptend function. It is the first value of r(3, n) to be greater than a googolplex. It starts with 70842519670086169864263801438..., and ends with ...5839. About 3210 digits in to this number, we find these digits: ...6950111111111111...*101 1s total*...1111111118195363078119....
1.6294043324... x 10995657055180967481723488710810833949177056029941963334338855462168341353507911292252707750506615682567
The factorial of a googol, sometimes referred to as the googolbang. It starts with 162940433245933737..., and ends with ...5616 followed by exactly 2.5*1099 - 18 zeroes.
2475912332475733472364471024200423............358932426146
2.4759123324... x 102929392929325571200070163461348562945302629498978182062432072821815998238017214192638690235643150485962835182238406135226103499082417
The triangrolplex. It is the first iterated triangle of 2 to be greater than 10 to the power of triangrol, and it has about 2.929*10^132 digits beginning with 2475912332475733472364471024200423..., and ending with ...358932426146.
2.3610226714... x 108072304726028225379382630397085399030071367921738743031867082828418414481568309149198911814701229483451981557574771156496457238535299087481244990261351116
4^4^4^4. Can also be expressed as 4^4^256, 2^2^513, 4^^4, or 4^^^2. Bowers used to call this number tritet, but he later reassigned that name to the much larger 4^^^^4 (seen in part 3), hence the name Tritet Jr.
1010^303
An ecetonplex.
102.88143726501021051... x 10^436
295 719 274 042 928 204 424 865 040 058 775 542 505 814 054 438 541 885 854 149 030 735 229 109 523 951 584 103 470 215 627 747 553 797 463 635 129 121 836 969 199 786 005 635 947 291 219 432 249 974 804 994 912 958 522 358 749 039 882 132 482 089 698 627 890 927 242 238 394 036 253 560 129 178 156 977 985 035 837 611 338 125 082 186 620 901 965 139 876 889 799 294 204 083 600 450 928 306 589 731 842 380 857 398 374 764 175 466 159 268 112 616 959 965 220 874 514 966 744 994 952 198 799 891 188 036 626 702 635 888 708 426 354 717 521 502 687 896 981 315 387 158 640 463 040 000, (2 to the power of the weak factorial of 1000) and thus the first number that is a square, cube, 4th power, 5th power, ..., 998th power, 999th power, and 1000th power. It starts with 11765156730407309953860743389954009625..., and ends with ...5509376.
3.9098490222... x 101124021466074751860097567522104789648012545442387518261576295420518517444766080791595055342613832148848657928846792570107753324167422010021177933700772606989124114395556249
831380979217553830172695027125136140707494294575475485321118531096366377975795248722471714194063487219461564568491620652987627661309480232956516340085351404053765203720536942043185
5146383193275981445894731731211119067826441631620760954270094664304695825570332511004312335248637332796979930683278729227794366058973791
2^3^4^5. This is an example of a number that can be expressed using power towers. It has more than 1.12*10488 digits beginning with 390984902223805103367261..., and ending with ...3472100352. It is much larger than 5^4^3^2 which has "just" 183,231 digits.
The Triple Exponential Range (10^10^500 ~ 10^10^10^80)
256^256^257 = 2^2^2059
= 39844342006.........365056
~ 101.992373902852*10^619
The second step in the calculation of the Mega. Can be expressed as 2vvv5 using down-arrow notation.
1010^1000
~ 10^(1.7718*10^1997)
2588568772411840194131602153234493556710295079477857120984192265232391789419880438906921921990316092705948991515485776046444825429596818069592027979684946307570829019934235587058964782020037324162761409406370304631006006530409780809946729268241385664636648641912737686541059272800551182722417047864174183908059599824896209575993796189207053873038187955600142079762745184357994797279737875654286166321802295860099158800366374549843738290997160174386317991999450594063920532823459585939820429244855525411860118417926631171779654659793784400589805837899013847271565554504341084459889511197310543346435630135802812930095615797602907232971854521270697049700951649924719937092125837323551211554870414993671041414677464208407554430043300186530390231964337897297668380866060195562956400040979303872526009423267268857697252474056885075564671228797634014731591716426588030909430219790056441909610707855080485779640350944209727501475184963379375681750980592735297610283090441817434192039935554462701881939443130632625602442747324700096861495216438083158096868200763242968319161648206544769058896421775705966984874767378351763642049808812344048530780627953430373753224986059653183547133795800568968483813955327073093092246118831867546968452807830777287523193646575447905175217058503452352430715087565242119026575975104684568694694282933761495934163898498895833912553641354411772371648959151433333377231917425854508829475647789957968940935660181063879064193907748179315923988850675337823763051948576639548553667740177679688563991034405660758518942653894203812227005139814690072143187517752829467362462871190565451799279855360842766034929939516723079856096501284206250470569504890716117396513914115620269557497773180198269629775587796242071327824364351971556771023719749743515764136906046632463733203007509819711888977867406538980331384029470004984193019899281555658286058672484322587527232862069965592949727958214753463780884938892181903933847487098166096452665106632745683143664200122860959024860772469439488504 , the smallest known power of 2 to contain a run of 1,998 consecutive 6s, hence the smallest "Goliath number". The block of sixes occurs within its last 3,000 digits.
5^^4 = 5^5^5^5 = 5^5^3,125
= 1111028808179974............368408203125
~ 101.33574*10^2184
55^5^5. Can also be expressed as 55^3125 or 5^^4. It starts with 1111028808179974... and ends with ...368408203125.
110128396785490937431723239464422714............010009008007006005004003002001
~ 101.1108... x10^2995
323281636769528800926351835023715983973984323619326165910831413485929742700614215858942141292106552658655336465357914729285203965080715619199116722509946605105626673348798836006162294182458797474912197648530006955831576558130503183566407157106681376566223204801235020125399185687545703740505109962601951819661206114099497774415669249849517161138917502311677484261152258180196442417263225035593038585881444310942065035095950968920888389323498161935201957798922364631858060372284385450391159744978925581552135811733071466716275536403692138744273025655311079042801587771071317707818903385967587979484369185094844251637825100180602091701300876876190228712613086437184900891973008415413477696907113884570850828598860696626410481432832844094100826116232464208915607276696040687112832527101783824480669487635858647253383141631348791228260728473906203725598228752389342927138278256534360590147021443638153883341336753743557373740258664286769593494932066218257550633397824448019376123474957267951100839969601433169299011802011353423919188601748382783035482235824456588469271162860703563827825008650136982644347755800437593829155230334977038413602212482422463082770749404366756571246832044509274708873551740261609250013164948050664827712410682109581186226279073091600705128891179623377611491193784719075593155134132119861544862985298953157046231119522631684692927840673684947765672758179718968195051448247021997594897806066825356698023148924431034439929167486436751376851275352433970244209116321685994874937454240000970260852611972489931273452282717090545149603043675870441461036998276359955007513619787255451780489258954265537968233037891487070908863055478972572155885891645274813928635609562785170231395436547214234661500453445623786721029033022048541732944548871220992681820063964305385904603945604977842198581605651064946875779905090889766773832652744222938527719657149024753614748515839492031568782193565415664874200229968066648896335517186105578256402313241340793141213794104216695907548602642379918936430864016649088748010144419406946003624367011037506878788757303930909519203355274584690593124735832595663826689493522300529514552691741876929779941519694758215851991668880927292010743110148171242562857668987845128822912305866712251501413711230794404236391257037956703998180856038614384774057293203055643604999066282108259591994285239555727967337561031646570430581567298707606370652940711613268758400882221281054963146345346057708680256470043815221248551070611875638429848732503207978402391534054221372415220031260793282280644916666252483076216355242870777936484322692691549040582101278345987944082069717503507453838119958141448884209889781617662644198843131526286262397115741973216919572875864790560514151386183619100907525011160547039478029143804667958049904149586286817594229383815283374141230187361581095697722247182288230817929105078090385316593009410672378202095683482818921143088889368689129227674208706841139417708342562906431362802786311036654466474123009080163880092407685548424371828934100.
The digits of this number end with a sort of countdown pattern: ...028027026025024023022021020019018017016015014013012011010009008007006005004003002001. The pattern occurs starting at the 2997th last digit. Multiplying this number by 998001 gives a number that ends with 2996 zeroes followed by a 1.
~ 109.0895*10^4815
2^2^16000. This number has something cursed in its first digits: 9221944678184914343981123467894434901218173123046909204778796241870670206677654750679179975...658443227136 (notice how it misses 5?)
1010^10,000
~ 102.413825645*10^15,937
9 * 87 * 65*4^(3*(2^1)))) This number begins with 6613158771793989045471703473..., and ends with ...852084414770288001024.
2^^6 = 2^2^65,536
= 212003872880821198488516469166227............3010305614986891826277507437428736
~ 106.031226*10^19,727
2^^6. Can also be expressed as 2^2^2^2^4, 2^2^2^16, 2^2^65536. It begins with 2120038728808211984885164691662274630835654230675372483625951752354414565561161..., and ends with ...3010305614986891826277507437428736. Notable for being the first power tower of 2s greater than googolplex.
6^^4 = 6^6^46,656
= 444623519023693469752.........3859420138438656
~ 102.06917*10^36,305
6^^4. Can also be expressed as 66^46656.
1010^100,000
A googolplexigong.
2^3^4^3^2 = 2^3^262,144
= 208813062276274558149621549031............805692076405844859168963428352
~ 109*10^125,073
23^4^3^2. An example of a number that can be expressed with a 5-height power tower, albeit a relatively small one. This number begins with 20881306227627455814962154903166664113427... and ends with ...805692076405844859168963428352.
6^5^4^3^2^1 = 6^5^262,144
= 11035602259176966321791453344753...............893380022607743740081787109376
~ 104.829*10^183,230
Exponential factorial of 6, the first exponential factorial bigger than googolplex. More digits are given below.
1103560225917696632179145334475344911419051187213486921746877175936136651431714552864199276654776541...042803106285328198624380944537003185235736096046992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
7^^4 = 7^7^7^7 = 7^7^823,543
= 783300523748005556540385409475............254036038182879357182733172343
~ 103.1*10^695,974
7^^4. Can also be expressed as 77^823543. This number begins with 7833005237480055565403854094754653082919... and ends with ...8511366058254036038182879357182733172343. Power towers of 7s actually have the cool property that each time you add a 7 to the tower, another two last digits become constant, not just one (this number has 6 convergent terminating digits).
1010^1,000,000
A millionduplex, the boundary between class-3 and class-4 numbers in Robert Munafo's idea of classes.
2^4^6^8 = 2^4^1,679,616 = 2^2^3,359,232
= 91232625227963370860086............795948302336
~1010^1,011,229
10^10^10,000,000
8^^4 = 8^8^16,777,216
= 64740329646697067997386625179............887920249826536946437619449856
~105.4*10^15,151,335
8^^4. Can also be expressed as 88^16,777,216, or 23*2^50331648.
~ 1010^20,538,505
33^3^2^2^2. This number begins with 145717531... and ends with ...8027.
~ 101.041392685158*10^43,046,721
1110^9^8. The decimal expansion of this number starts 2373570079623504840997604286196... and ends ...8446 followed by 43,046,721 zeroes and then a 1. The first 6 digits are all prime.
1010^100,000,000
A googolplexibong.
6^7^8^9 = 6^7^134,217,728
= 25645487552806817355151.........204000256
~ 10^(5.7415*10^113,427,138)
Giganika
488^2^1,000,000,000
= ? ......... 8816
~104.085*10^369,693,099
9^^4. The largest number that can be made with 4 digits and the basic mathematical operators. This number begins with 21419832947968056113333.... and ends with ...401045865289.
1010^1,000,000,000
A billiduplexion.
2^2^2^2^5 = 2^2^2^32 = 2^2^4294967296
= 315921269337233843004184822325...............78717272631317364736
~ 109.3418*10^1,292,913,985
The largest number I have directly calculated the leading digits of. It starts with 315921269337233843004184822... (?) and ends with ...78717272631317364736. I'm not even 100% sure if the result for this number's first digits is correct, as Mathematica may not be reliable to that many digits.
4^4^4^4^2 = 4^4^4^16 = 4^4^4294967296
= ? ............318192949300735263328903888896
~ 105.78*10^2,585,827,972
44^4^4^2. Can also be expressed as 2^2^8589934593.
The last 100 digits are
...5586637447420842934283811065743145570037983353602268026980849968305882318192949300735263328903888896. It may be possible to calculate this number's first digits; I will try if I can find (or write) another program that can do large calculations.
~ 108.038723426*10^3,327,237,905
3vvv4, or 33^6973568822. The last 4 digits are ...5683.
1010^10,000,000,000
10^^4, a tetralogue. Notable for being about as far as we can currently go in calculating the leading digits of large powers (however, with more dedicated resources, the limit could be extended a bit further, see 3^^5 on the next page).