# Largest known prime number

The largest known prime number is 2$^{136,279,841}$ − 1, a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).
A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. As of October 2024, the seven largest known primes are Mersenne primes. The last eighteen record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2$^k$ − 1 is simply k ones.
Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not the case.

## Current record

The record is currently held by 2$^{136,279,841}$ − 1 with 41,024,320 digits, found by GIMPS on October 12, 2024. The first and last 120 digits of its value are:
   881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874 ...
   (41,024,080 digits skipped)
   ... 852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

## Prizes

There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes. A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize. In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF. Time called this prime the 29th top invention of 2008.
Both of these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further US$250,000 prize is offered for the first prime with at least one billion digits.
GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.

## History

The following table lists the progression of the largest known prime number in ascending order. Here M$_p$ = 2$^p$ − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M$_{19}$ = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.
GIMPS volunteers found the sixteen latest records, all of them Mersenne primes. They were found on ordinary personal computers until the most recent one, found by ex-Nvidia employee Luke Durant using a network of thousands of dedicated graphics processing units (GPUs). Durant spent almost exactly one year and approximately US$2 million of his personal money on the hunt. This achievement marks the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs), ushering in a new era in prime number searches.
| Number | Digits | First 120 digits | Last 120 digits | Year found | Discoverer |
| M$_{13}$ | 4 | 8191 | 8191 | 1456 | Anonymous |
| M$_{17}$ | 6 | 131071 | 131071 | 1588 | Pietro Cataldi |
| M$_{19}$ | 6 | 524287 | 524287 | 1588 | Pietro Cataldi |
| ${\tfrac {2^{32}+1}{641}}$ | 7 | 6700417 | 6700417 | 1732 | Leonhard Euler? Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 2$^{32}$ + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it. |
| M$_{31}$ | 10 | 2147483647 | 2147483647 | 1772 | Leonhard Euler |
| ${\tfrac {10^{18}+1}{1000001}}$ | 12 | 999999000001 | 999999000001 | 1851 | Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record. |
| ${\tfrac {2^{64}+1}{274177}}$ | 14 | 67280421310721 | 67280421310721 | 1855 | Thomas Clausen (but no proof was provided). |
| M$_{127}$ | 39 | 17014118346046923173 1687303715884105727 | 1701411834604692317 31687303715884105727 | 1876 | Édouard Lucas |
| ${\tfrac {2^{148}+1}{17}}$ | 44 | 20988936657440586486 15126425661022259386 3921 | 2098 89366574405864861512 64256610222593863921 | 1951 | Aimé Ferrier with a mechanical calculator; the largest record not set by computer. |
| 180×(M$_{127}$)$^2$+1 | 79 | 52106440156792287940 60694325390955853335 89848390805645835218 3851018372555735221 | 5210644015679228794 06069432539095585333 58984839080564583521 83851018372555735221 | 1951 | J. C. P. Miller & D. J. Wheeler Using Cambridge's EDSAC computer |
| M$_{521}$ | 157 | 68647976601306097149 81900799081393217269 43530014330540939446 34591855431833976560 52122559640661454554 97729631139148085803 | 26943530014330540939 44634591855431833976 56052122559640661454 55497729631139148085 80371219879997166438 12574028291115057151 | 1952 | Raphael M. Robinson |
| M$_{607}$ | 183 | 53113799281676709868 95882065524686273295 93117727031923199444 13820040355986085224 27391625022652292856 68889329486246501015 | 20040355986085224273 91625022652292856688 89329486246501015346 57933765270723940951 99787665873519438312 70835393219031728127 | 1952 | Raphael M. Robinson |
| M$_{1279}$ | 386 | 10407932194664399081 92524032736408553861 52622472667048053191 12350403608059673360 29801223944173232418 48424216139542810077 | 82853841658502825560 46662248318909188018 47068222203140521026 69843548873295802887 80508697361869007147 20710555703168729087 | 1952 | Raphael M. Robinson |
| M$_{2203}$ | 664 | 14759799152141802350 84898622737381736312 06614533316977514777 12164785702978780789 49377407337049389289 38274850753149648047 | 51945258754287534997 65585726702296339625 75212637477897785501 55264652260998886991 40135404838098656812 50419497686697771007 | 1952 | Raphael M. Robinson |
| M$_{2281}$ | 687 | 44608755718375842957 11517064021018098862 08632412859901111991 21996340468579282047 33691125452690039890 26153245931124316702 | 95491713975879606122 38033935373810346664 94402951052059047968 69325538864793044092 51041868170096401717 64133172418132836351 | 1952 | Raphael M. Robinson |
| M$_{3217}$ | 969 | 25911708601320262777 62467679224415309418 18887553125427303974 92316187401926658636 20862012095168004834 06550695241733194177 | 59459944433523118828 01236604062624686092 12150349937584782292 23714433962885848593 82157388212323936870 46160677362909315071 | 1957 | Hans Riesel |
| M$_{4423}$ | 1,332 | 28554254222827961390 15635661021640083261 64238644702889199247 45660228440039060065 38759545715055398432 39754513915896150297 | 82106760176875097786 61004600146021384084 48021225053689054793 74200309572209673295 47507217181155318713 10231057902608580607 | 1961 | Alexander Hurwitz |
| M$_{9689}$ | 2,917 | 47822027880546120295 28392986600059097414 97172402236500851334 51099183789509426629 70278927686112707894 58682472098152425631 | 96502507081973046642 28261056975105642897 98951182192885976352 22905389894873761464 21399109115358645058 18992696826225754111 | 1963 | Donald B. Gillies |
| M$_{9941}$ | 2,993 | 34608828249085121524 29603957674133167226 28668900238547790489 28344500622080983411 44643643755441537075 33664486747635050186 | 85925083476189478888 95525278984009881962 00014868575640233136 50914562812719135485 82750839078914699790 19426224883789463551 | 1963 | Donald B. Gillies |
| M$_{11213}$ | 3,376 | 28141120136973731333 93152975842584191818 66238201360078789241 93493455151766822763 13810715094745633257 07419878930853507153 | 87566914032072497856 85867185275866024396 02335283513944980064 32703027810422414497 18836805416897847962 67391476087696392191 | 1963 | Donald B. Gillies |
| M$_{19937}$ | 6,002 | 43154247973881626480 55235516337919839053 93504322671150516525 05414033306801376580 91130451362931858466 55452699382576488353 | 60727895549548774214 07535706212171982521 92978869786916734625 61843017545490386411 15854295045699209056 36741539030968041471 | 1971 | Bryant Tuckerman |
| M$_{21701}$ | 6,533 | 44867916611904333479 49514103615917787272 09023729388613010364 80447512785609158053 63716201839592018310 86891496139730355336 | 33369896693354436162 93913110417309565016 94662754558875644345 19126927960069355180 92719564502642940928 57410828353511882751 | 1978 | Laura A. Nickel and Landon Curt Noll |
| M$_{23209}$ | 6,987 | 40287411577898877818 18733290715917677224 38506891622420041029 96357869459524088740 08676398614614665371 03833299413586592359 | 49990785611757500951 57465578625397647565 74427752110896827606 78602528203915287605 50508545118172938900 36743355523779264511 | 1979 | Landon Curt Noll |
| M$_{44497}$ | 13,395 | 85450982430363380319 33007053184030365099 01591304021058343269 25828229006478216763 58562005000144576458 61481315295253223674 | 19107442963978359909 48993204100398635759 46472558059877105808 94247177392297739634 54976377895623405368 44867686961011228671 | 1979 | David Slowinski and Harry L. Nelson |
| M$_{86243}$ | 25,962 | 53692799550275632152 23382779929453006110 20994042124005915678 63944335346298210347 98964395551413140596 01329696868637207994 | 57351862519228939958 84693761059056977054 15089600178032945914 35320137691545632232 02509608679061957196 99857021709433438207 | 1982 | David Slowinski |
| M$_{132049}$ | 39,751 | 51274027626932072381 27857636203402218800 46586227069926831240 38418582312743056203 61077749499092908732 12555709320045159618 | 89256188390637660219 36832367367308227116 78956149432532644153 24079640048510932988 33786316447035663398 52138578455730061311 | 1983 | David Slowinski |
| M$_{216091}$ | 65,050 | 74609310306466134368 73395794005114895402 28754084977328805113 30497779366272527096 87806643956351409557 30008364494154882757 | 41796441616213691597 66435268814054587246 91315195450691201831 18538411805217750684 69327867645141118776 91336204103815528447 | 1985 | David Slowinski |
| 391581×2$^{216193}$−1 | 65,087 | 14814063237640662751 89896116681502152616 14869061837067878963 23169460093384999355 40035564748752481896 29946106929509682950 | 82819868449333023401 04392759176586303336 22389718952919899041 01638046268529515895 76118449880787230436 89626791836387377151 | 1989 | A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello. Largest non-Mersenne prime that was the largest known prime when it was discovered. |
| M$_{756839}$ | 227,832 | 17413590682008709732 51635992459033278907 79363690507030974654 73553838272156206625 76319147974364224616 10635130071368293660 | 19619724789014565809 44396409267168409183 49113692649241768590 51134272012692706848 76804040558133428809 02603793328544677887 | 1992 | David Slowinski and Paul Gage |
| M$_{859433}$ | 258,716 | 12949812560420764966 65334852555620733841 62019917416569370190 66267567814724084952 96919893191078354681 55567280151644798137 | 70366138430104674404 17291687756716831654 19536906002518061544 66211087607689521384 87432526245965721589 02414267243500142591 | 1994 | David Slowinski and Paul Gage |
| M$_{1257787}$ | 378,632 | 41224577362142867472 53232184669789600527 87185654659469380413 20489580405544505611 40313191552792105979 05669363277683158359 | 92352317328348412624 08558666851703702032 47995651850069878600 72644421009952433369 54631641051358552671 31257188976089366527 | 1996 | David Slowinski and Paul Gage |
| M$_{1398269}$ | 420,921 | 81471756441257307514 26772643891354260153 13783085022271032114 51048469938030899616 08340980239948586278 86398792156198534051 | 70112944662406744358 62878919205295726467 35633955407734562739 68427460950363262807 77790674776834625319 85532025868451315711 | 1996 | GIMPS, Joel Armengaud |
| M$_{2976221}$ | 895,932 | 62334007624857864988 60414411708927450502 70498680527705762010 44980837228500531612 87552386408711765558 35347026816848251160 | 12689188858968205493 08475288306533381326 50949313652525946734 18989311375605582078 15564860085353060451 76506256743729201151 | 1997 | GIMPS, Gordon Spence |
| M$_{3021377}$ | 909,526 | 12741168303009336743 35542151767349261473 65409710390533367899 30486889243847834725 96446989025955854374 97756265138125839679 | 47478189918377204959 69880392336860732039 11214513449538158982 93606342963753971823 36558874582102617702 25422631973024694271 | 1998 | GIMPS, Roland Clarkson |
| M$_{6972593}$ | 2,098,960 | 43707574412708137883 33232912069460708676 24770574851606631018 13181519232482250706 53865555856672485830 59003027082699320939 | 73675389080631004085 08543235704913317476 87718276359853562553 41815592459312082762 45050174988400346151 35366526142924193791 | 1999 | GIMPS, Nayan Hajratwala |
| M$_{13466917}$ | 4,053,946 | 92494773800670132224 77583825476640519253 54401079958299021030 93608029565658055961 00476131215557305846 49024542650476541902 | 22828849378011781756 76448390574570798287 48568541687729337577 30752297148385814257 76644015462093334911 30073855470256259071 | 2001 | GIMPS, Michael Cameron |
| M$_{20996011}$ | 6,320,430 | 12597689545033010502 04943095748243114559 93416085351835952254 67012565498768908351 56022124009680282853 61325441271583233254 | 53656018582721448133 13954215503264848667 10969127787170820477 53340930097294847523 19834716766530781632 94714065762855682047 | 2003 | GIMPS, Michael Shafer |
| M$_{24036583}$ | 7,235,733 | 29941042940415717208 90489263404469382573 67722975418473547677 34860009764022110074 10262658651099123208 58493344156415212635 | 97367931835649549332 62413429503748554259 55207718464378183256 42314252685868703980 05560312691184129150 67436921882733969407 | 2004 | GIMPS, Josh Findley |
| M$_{25964951}$ | 7,816,230 | 12216463006127794810 77539640312884392673 61424223075246409537 66046996455809056861 56907748512690404182 46405468474387100505 | 82841605918218299877 77039869777444372767 13026360619053009303 03992810433168520775 07113305351596265166 98933257280577077247 | 2005 | GIMPS, Martin Nowak |
| M$_{30402457}$ | 9,152,052 | 31541647561884608093 63030286645451701265 19656262323870316323 71079513538744900693 46209438629475170296 63623614229944506869 | 29904518450254170958 38942393049606751896 53422547853529862010 43713583091577749950 02748822185508467086 11134297411652943871 | 2005 | GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone |
| M$_{32582657}$ | 9,808,358 | 12457502601536945540 08555015747995031227 95985151151842843670 47566259111523599739 73805597596066168459 39100419886882111308 | 72660495893732258251 20726126214431145356 41869584273577446330 45746582133321244573 71046356920000926590 11752880154053967871 | 2006 | GIMPS, Curtis Cooper and Steven Boone |
| M$_{43112609}$ | 12,978,189 | 31647026933025592314 34537239493375160541 06188475264644140304 17673281124749306936 86920431851216118378 56726816539985465097 | 15927979190839813022 33048240831190931959 98014562456347941202 19590092807967072944 79216164918874782657 80022181166697152511 | 2008 | GIMPS, Edson Smith |
| M$_{57885161}$ | 17,425,170 | 58188726623224644217 51002121132323686363 70852325421589325781 70448058449276170744 23164282813494233769 42979071335489886655 | 19696440089898189117 97158303938275980625 06665259086044516822 49493774541094283332 30952037056456587257 46141988071724285951 | 2013 | GIMPS, Curtis Cooper |
| M$_{74207281}$ | 22,338,618 | 30037641808460618205 29860983591660500568 75863030301484843941 69334554772321906799 42968936553007726883 20448214882399426727 | 71777401476291246211 36468794258014451073 93100212927181629335 93149423901821387921 76711649562871904986 87010073391086436351 | 2016 | GIMPS, Curtis Cooper |
| M$_{77232917}$ | 23,249,425 | 46733318335923109998 83355855611155212513 21102817714495798582 33859356792348052117 72074843110997402088 49621368090038049317 | 28537600451878605540 22233766729256792821 31965467343395945397 37047636927989462799 99396146592173711365 82730618069762179071 | 2017 | GIMPS, Jonathan Pace |
| M$_{82589933}$ | 24,862,048 | 14889444574204132554 78064584723979166030 26273992795324185271 28942521323936106447 53103099711321803371 74752834401423587560 | 06210755794795829753 15952088071926936765 21782184472526640076 91211435530831196948 76337664578236950740 37951210325217902591 | 2018 | GIMPS, Patrick Laroche |
| M$_{136279841}$ | 41,024,320 | 88169432750383326555 39391003781173589712 07354509066041067156 37641242263069475684 14417259903477232831 08837509739959776874 | 85280651793145941256 79575682842282881240 96109707961148305849 34976608576417071506 04094045096221046655 55076706219486871551 | 2024 | GIMPS, Luke Durant |

## Twenty largest

A list of the 5,000 largest known primes is maintained by the PrimePages, of which the twenty largest are listed below.
| Rank | Number | Discovered | Digits | First 120 digits | Last 120 digits | Form |
| 1 | 2$^{136279841}$ − 1 | 2024-10-12 | 41,024,320 | 88169432750383326555 39391003781173589712 07354509066041067156 37641242263069475684 14417259903477232831 08837509739959776874 | 85280651793145941256 79575682842282881240 96109707961148305849 34976608576417071506 04094045096221046655 55076706219486871551 | Mersenne |
| 2 | 2$^{82589933}$ − 1 | 2018-12-07 | 24,862,048 | 14889444574204132554 78064584723979166030 26273992795324185271 28942521323936106447 53103099711321803371 74752834401423587560 | 06210755794795829753 15952088071926936765 21782184472526640076 91211435530831196948 76337664578236950740 37951210325217902591 | Mersenne |
| 3 | 2$^{77232917}$ − 1 | 2017-12-26 | 23,249,425 | 46733318335923109998 83355855611155212513 21102817714495798582 33859356792348052117 72074843110997402088 49621368090038049317 | 28537600451878605540 22233766729256792821 31965467343395945397 37047636927989462799 99396146592173711365 82730618069762179071 | Mersenne |
| 4 | 2$^{74207281}$ − 1 | 2016-01-07 | 22,338,618 | 30037641808460618205 29860983591660500568 75863030301484843941 69334554772321906799 42968936553007726883 20448214882399426727 | 71777401476291246211 36468794258014451073 93100212927181629335 93149423901821387921 76711649562871904986 87010073391086436351 | Mersenne |
| 5 | 2$^{57885161}$ − 1 | 2013-01-25 | 17,425,170 | 58188726623224644217 51002121132323686363 70852325421589325781 70448058449276170744 23164282813494233769 42979071335489886655 | 19696440089898189117 97158303938275980625 06665259086044516822 49493774541094283332 30952037056456587257 46141988071724285951 | Mersenne |
| 6 | 2$^{43112609}$ − 1 | 2008-08-23 | 12,978,189 | 31647026933025592314 34537239493375160541 06188475264644140304 17673281124749306936 86920431851216118378 56726816539985465097 | 15927979190839813022 33048240831190931959 98014562456347941202 19590092807967072944 79216164918874782657 80022181166697152511 | Mersenne |
| 7 | 2$^{42643801}$ − 1 | 2009-06-04 | 12,837,064 | 16987351645274162247 02898707511764713591 03325776997255365512 60020505373109218621 22599292756037678425 64017793851584510263 | 89793266835248591744 64060649185927134914 73117475647591955485 69867927456113537511 49133460978428956443 84101954765562314751 | Mersenne |
| 8 | Φ$_3$(−516693$^{1048576}$) | 2023-10-02 | 11,981,518 | 13402906796489222357 52246822000881801252 41118044574855268822 40787049468713337605 50197597945996229191 43176765531862533944 | 45102449632978070416 89341970562017911020 84113168162771694298 54415779073874568943 91416059782334617095 67178301964288000001 | Generalized unique |
| 9 | Φ$_3$(−465859$^{1048576}$) | 2023-05-31 | 11,887,192 | 17395442163066427324 04095947530927014429 23721230469791611973 13180378592661492867 58297267063261966785 92548252101237137788 | 85278914675748208502 55226473801289095503 68054401147310815004 88562918084974370698 95163405490252252372 63508838734878474241 | Generalized unique |
| 10 | 2$^{37156667}$ − 1 | 2008-09-06 | 11,185,272 | 20225440689097733553 41881522631568299468 46602582743182989551 05736054751457975812 50846721390095896345 30142096674488997709 | 14728787551899048539 16991622232001005966 66765048100145151363 48394299744493358135 21893866570487429610 21340265022308220927 | Mersenne |
| 11 | 2$^{32582657}$ − 1 | 2006-09-04 | 9,808,358 | 12457502601536945540 08555015747995031227 95985151151842843670 47566259111523599739 73805597596066168459 39100419886882111308 | 72660495893732258251 20726126214431145356 41869584273577446330 45746582133321244573 71046356920000926590 11752880154053967871 | Mersenne |
| 12 | 10223 × 2$^{31172165}$ + 1 | 2016-10-31 | 9,383,761 | 50625026920996343077 76282032439067604835 90666966114515920950 45640633412043430359 88815895056171116175 51873728066666193155 | 91892134918826938976 55779680218334368800 88050529917153697492 60915967379870147035 24878105802550394137 86610918915347316737 | Proth |
| 13 | 2$^{30402457}$ − 1 | 2005-12-15 | 9,152,052 | 31541647561884608093 63030286645451701265 19656262323870316323 71079513538744900693 46209438629475170296 63623614229944506869 | 29904518450254170958 38942393049606751896 53422547853529862010 43713583091577749950 02748822185508467086 11134297411652943871 | Mersenne |
| 14 | 4 × 5$^{11786358}$ + 1 | 2024-10-01 | 8,238,312 | 20156998396261662175 28359889367930265681 50456335975784718728 38256327105334814872 69679155318722963484 77606435567527820548 | 96709057831042893049 21132697813111998064 54292933825858091969 62763705692893998269 73241403038628050126 13534927368164062501 | Generalized Proth |
| 15 | 2$^{25964951}$ − 1 | 2005-02-18 | 7,816,230 | 12216463006127794810 77539640312884392673 61424223075246409537 66046996455809056861 56907748512690404182 46405468474387100505 | 82841605918218299877 77039869777444372767 13026360619053009303 03992810433168520775 07113305351596265166 98933257280577077247 | Mersenne |
| 16 | 69 × 2$^{24612729}$ − 1 | 2024-08-13 | 7,409,102 | 34913857494942645537 77528193541070245743 51335040706255350040 22702450446821700067 28827887453950698207 63928288182629713589 | 05695137582497488595 79121604235601653208 59352298128248331223 49160290220193535509 71657492602305174873 93807281434214268927 | Riesel |
| 17 | 2$^{24036583}$ − 1 | 2004-05-15 | 7,235,733 | 29941042940415717208 90489263404469382573 67722975418473547677 34860009764022110074 10262658651099123208 58493344156415212635 | 97367931835649549332 62413429503748554259 55207718464378183256 42314252685868703980 05560312691184129150 67436921882733969407 | Mersenne |
| 18 | 107347 × 2$^{23427517}$ − 1 | 2024-08-04 | 7,052,391 | 23535192646535179116 38946094063474658764 68924164622481357963 62977099077527159960 22049070416163357350 57403900382750381230 | 94089016871571688757 09838699794575028002 54586750694611329151 02052885568916854511 29696212376296097359 46366182881423785983 | Riesel |
| 19 | 3843236$^{1048576}$ + 1 | 2024-12-22 | 6,904,556 | 19856879544723191042 77924219146224458915 90804518885059407859 80294632609085741518 21088549676823013678 37154935540090480218 | 79027381231135136406 25000778200220184245 00224553095520553849 95924225949910453582 41080785482138415590 96566069268616052737 | Generalized Fermat |
| 20 | 3 × 2$^{22103376}$ − 1 | 2024-09-30 | 6,653,780 | 45557575201836797391 77924694750863521479 98028844478065239917 78652383915500660790 31585797936442894001 65356897435998223877 | 92420137288192690503 91322027063289672125 39967518049052118055 78416884567426783748 98800257594371326293 13552387699174801407 | Thabit |