Largest known prime number
From Wikipedia, the free encyclopedia
The largest known prime number as of 2026[update] is 2136,279,841 − 1, a number that 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 researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).[1][2]
A prime number is a natural number greater than 1 with no divisors other than 1 and itself. Euclid's theorem proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; 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 be verified by a specialized primality test that is faster than the general one. As of October 2024[update], the seven largest known primes are Mersenne primes.[3] The last 18 record primes were Mersenne primes.[4][5] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.[6]
Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not true. While large primes with hundreds of digits are indeed used for cryptography, primes with millions of digits are not.[7][8]
Current record
The record is held by 2136,279,841 − 1 with 41,024,320 digits, found by GIMPS on October 12, 2024.[1] Its first and last 120 digits are:[10]
881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874...
[41,024,080 digits skipped]
...852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551
Prizes
The Electronic Frontier Foundation (EFF) offers several prizes for record primes.[11] A prime with one million digits was found in 1999, earning the discoverer a $50,000 prize.[12] In 2008, a ten-million-digit prime won a $100,000 prize and a Cooperative Computing Award from the EFF.[11] Time called this prime the 29th top invention of 2008.[13]
Both 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 $150,000 prize for the first prime of over 100 million digits with the winning participant. A $250,000 prize awaits the first prime with one billion digits.[11]
GIMPS also offers a $3,000 research discovery award for participants who discover a new Mersenne prime of fewer than 100 million digits.[14]
History
The following table lists the progression of the largest known prime number in ascending order.[4] Here Mp = 2p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder was M19 = 524,287, which held the record for 184 years.
The primes up to and including were found without a computer, while the primes starting with 180×(M127)2+1 were found using computers.
GIMPS volunteers found the 16 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).[1] Durant spent about a year and $2 million on the hunt.[15] This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).[16][17]
, 
...| Number | Digits | Year found | Discoverer |
|---|---|---|---|
| M17 | 6 | 1588 | Pietro Cataldi |
| M19 | 6 | 1588 | Pietro Cataldi |
| M31 | 10 | 1772 | Leonhard Euler |
| 13 | 1867 | Fortuné Landry | |
| M127 | 39 | 1876 | Édouard Lucas |
| 44 | 1951 | Aimé Ferrier, with a mechanical calculator. The largest record not set by computer. | |
| 180×(M127)2+1 | 79 | 1951 | J. C. P. Miller & D. J. Wheeler[18] using Cambridge's EDSAC computer |
| M521 | 157 | 1952 | Raphael M. Robinson |
| M607 | 183 | 1952 | Raphael M. Robinson |
| M1279 | 386 | 1952 | Raphael M. Robinson |
| M2203 | 664 | 1952 | Raphael M. Robinson |
| M2281 | 687 | 1952 | Raphael M. Robinson |
| M3217 | 969 | 1957 | Hans Riesel |
| M4423 | 1,332 | 1961 | Alexander Hurwitz |
| M9689 | 2,917 | 1963 | Donald B. Gillies |
| M9941 | 2,993 | 1963 | Donald B. Gillies |
| M11213 | 3,376 | 1963 | Donald B. Gillies |
| M19937 | 6,002 | 1971 | Bryant Tuckerman |
| M21701 | 6,533 | 1978 | Laura A. Nickel and Landon Curt Noll[19] |
| M23209 | 6,987 | 1979 | Landon Curt Noll[19] |
| M44497 | 13,395 | 1979 | David Slowinski and Harry L. Nelson[19] |
| M86243 | 25,962 | 1982 | David Slowinski[19] |
| M132049 | 39,751 | 1983 | David Slowinski[19] |
| M216091 | 65,050 | 1985 | David Slowinski[19] |
| 391581×2216193−1 | 65,087 | 1989 | The "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[20][21] Largest non-Mersenne prime that was the largest known prime when it was discovered. |
| M756839 | 227,832 | 1992 | David Slowinski and Paul Gage[19] |
| M859433 | 258,716 | 1994 | David Slowinski and Paul Gage[19] |
| M1257787 | 378,632 | 1996 | David Slowinski and Paul Gage[19] |
| M1398269 | 420,921 | 1996 | GIMPS, Joel Armengaud |
| M2976221 | 895,932 | 1997 | GIMPS, Gordon Spence |
| M3021377 | 909,526 | 1998 | GIMPS, Roland Clarkson |
| M6972593 | 2,098,960 | 1999 | GIMPS, Nayan Hajratwala |
| M13466917 | 4,053,946 | 2001 | GIMPS, Michael Cameron |
| M20996011 | 6,320,430 | 2003 | GIMPS, Michael Shafer |
| M24036583 | 7,235,733 | 2004 | GIMPS, Josh Findley |
| M25964951 | 7,816,230 | 2005 | GIMPS, Martin Nowak |
| M30402457 | 9,152,052 | 2005 | GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone |
| M32582657 | 9,808,358 | 2006 | GIMPS, Curtis Cooper and Steven Boone |
| M43112609 | 12,978,189 | 2008 | GIMPS, Edson Smith |
| M57885161 | 17,425,170 | 2013 | GIMPS, Curtis Cooper |
| M74207281 | 22,338,618 | 2016 | GIMPS, Curtis Cooper |
| M77232917 | 23,249,425 | 2017 | GIMPS, Jonathan Pace |
| M82589933 | 24,862,048 | 2018 | GIMPS, Patrick Laroche |
| M136279841 | 41,024,320 | 2024 | GIMPS, Luke Durant |
Twenty largest
A list of the 5,000 largest known primes is maintained by the PrimePages,[22] of which the largest 20 are listed below.[23]
| Rank | Number | Discovered | Digits | Form | Ref. |
|---|---|---|---|---|---|
| 1 | 2136279841 − 1 | 2024-10-12 | 41,024,320 | Mersenne | [1] |
| 2 | 282589933 − 1 | 2018-12-07 | 24,862,048 | Mersenne | [24] |
| 3 | 277232917 − 1 | 2017-12-26 | 23,249,425 | Mersenne | [25] |
| 4 | 274207281 − 1 | 2016-01-07 | 22,338,618 | Mersenne | [26] |
| 5 | 257885161 − 1 | 2013-01-25 | 17,425,170 | Mersenne | [27] |
| 6 | 25241902097152 + 1 | 2025-10-12 | 13,426,224 | Generalized Fermat | [28] |
| 7 | 243112609 − 1 | 2008-08-23 | 12,978,189 | Mersenne | [29] |
| 8 | 242643801 − 1 | 2009-06-04 | 12,837,064 | Mersenne | [30] |
| 9 | Φ3(−5166931048576) | 2023-10-02 | 11,981,518 | Generalized unique | [31] |
| 10 | Φ3(−4658591048576) | 2023-05-31 | 11,887,192 | Generalized unique | [32] |
| 11 | 237156667 − 1 | 2008-09-06 | 11,185,272 | Mersenne | [29] |
| 12 | 232582657 − 1 | 2006-09-04 | 9,808,358 | Mersenne | [33] |
| 13 | 10223 × 231172165 + 1 | 2016-10-31 | 9,383,761 | Proth | [34] |
| 14 | 230402457 − 1 | 2005-12-15 | 9,152,052 | Mersenne | [35] |
| 15 | 4 × 511786358 + 1 | 2024-10-01 | 8,238,312 | Generalized Proth | [36] |
| 16 | 225964951 − 1 | 2005-02-18 | 7,816,230 | Mersenne | [37] |
| 17 | 4052186 × 694052186 + 1 | 2025-04-17 | 7,451,366 | Generalized Cullen | [38] |
| 18 | 69 × 224612729 − 1 | 2024-08-13 | 7,409,102 | Riesel | [39] |
| 19 | 224036583 − 1 | 2004-05-15 | 7,235,733 | Mersenne | [40] |
| 20 | 53362841048576 + 1 | 2025-11-02 | 7,054,022 | Generalized Fermat | [41] |