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The largest known prime number is 2136,279,841 − 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).[1][2]
A plot of the number of digits in the largest known prime by year, since the electronic computer. The vertical scale is logarithmic.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 utilize 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 eighteen 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 the case.[7][8]
The standard format of 50 lines per page, 75 characters per line would require 10,940 single-sided pages to print this prime number, or approximately 22 reams of paper. (15 are shown here.)[9]The record is currently held by 2136,279,841 − 1 with 41,024,320 digits, found by GIMPS on October 12, 2024.[1] The first and last 120 digits of its value are:[10]
881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874 ...
(41,024,080 digits skipped)
... 852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551
There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[11] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[12] In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[11] Time called this prime the 29th top invention of 2008.[13]
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.[11]
GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[14]
Commemorative postmark used by the UIUC Math Department after proving that M11213 is primeThe 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 known was M19 = 524,287, which was the largest known prime for 144 years.
The primes up to and including 2 148 + 1 17 {\displaystyle {\tfrac {2^{148}+1}{17}}} are found without a computer, while the primes starting with 180×(M127)2+1 are found using computers.
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).[1] Durant spent about one year and US$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]
Largest known prime by year[4] Number Digits Year found Discoverer M17 6 1588 Pietro Cataldi M19 6 1588 Pietro Cataldi M31 10 1772 Leonhard Euler M 59 179951 {\displaystyle {\mathsf {\tfrac {M_{59}}{179951}}}} 13 1867 Fortuné Landry M127 39 1876 Édouard Lucas 2 148 + 1 17 {\displaystyle {\mathsf {\tfrac {2^{148}+1}{17}}}} 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]A list of the 5,000 largest known primes is maintained by the PrimePages,[22] of which the twenty largest 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 243112609 − 1 2008-08-23 12,978,189 Mersenne [28] 7 242643801 − 1 2009-06-04 12,837,064 Mersenne [29] 8 Φ3(−5166931048576) 2023-10-02 11,981,518 Generalized unique [30] 9 Φ3(−4658591048576) 2023-05-31 11,887,192 Generalized unique [31] 10 237156667 − 1 2008-09-06 11,185,272 Mersenne [28] 11 232582657 − 1 2006-09-04 9,808,358 Mersenne [32] 12 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [33] 13 230402457 − 1 2005-12-15 9,152,052 Mersenne [34] 14 4 × 511786358 + 1 2024-10-01 8,238,312 Generalized Proth [35] 15 225964951 − 1 2005-02-18 7,816,230 Mersenne [36] 16 4052186 × 694052186 + 1 2025-04-17 7,451,366 Generalized Cullen [37] 17 69 × 224612729 − 1 2024-08-13 7,409,102 Riesel [38] 18 224036583 − 1 2004-05-15 7,235,733 Mersenne [39] 19 107347 × 223427517 − 1 2024-08-04 7,052,391 Riesel [40] 20 3 × 223157875 − 1 2025-06-25 6,971,216 Thabit [41]An interesting side note is about the binary representations of those numbers...
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