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).^[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 2^k − 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]

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.^[1] The first and last 120 digits of its value are:^[9]

881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874 ...

(41,024,080 digits skipped)

... 852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

Prizes

There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.^[10] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.^[11] In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.^[10] Time called this prime the 29th top invention of 2008.^[12]

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.^[10]

GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.^[13]

History

Commemorative postmark used by the UIUC Math Department after proving that M₁₁₂₁₃ is prime

The following table lists the progression of the largest known prime number in ascending order.^[4] 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 M₁₉ = 524,287, which was the largest known prime for 144 years.

The primes up to and including ${\displaystyle {\tfrac {2^{148}+1}{17}}}$ 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.^[14] This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).^[15][16]

Number	Digits	Year found	Discoverer
M13	4	1456	Anonymous
M17	6	1588	Pietro Cataldi
M19	6	1588	Pietro Cataldi
${\displaystyle {\tfrac {2^{32}+1}{641}}}$	7	1732	Leonhard Euler? Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.[17]
M31	10	1772	Leonhard Euler
${\displaystyle {\tfrac {10^{18}+1}{1000001}}}$	12	1851	Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record.
${\displaystyle {\tfrac {2^{64}+1}{274177}}}$	14	1855	Thomas Clausen (but no proof was provided).
M127	39	1876	Édouard Lucas
${\displaystyle {\tfrac {2^{148}+1}{17}}}$	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	A group, "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 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	38432361048576 + 1	2024-12-17	6,904,556	Generalized Fermat	[41]

See also

• List of largest known primes and probable primes