The largest known prime number is , 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).

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. , the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes.The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.

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.

Current record

The record is held by with 41,024,320 digits, found by GIMPS on October 12, 2024. Its first and last 120 digits are:

Prizes The Electronic Frontier Foundation (EFF) offers several prizes for record primes. A prime with one million digits was found in 1999, earning the discoverer a $50,000 prize.Electronic Frontier Foundation, Big Prime Nets Big Prize. In 2008, a ten-million-digit prime won a $100,000 prize and a Cooperative Computing Award from the EFF. Time called this prime the 29th top invention of 2008.

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.

GIMPS also offers a $3,000 research discovery award for participants who discover a new Mersenne prime of fewer than 100 million digits.

History

The following table lists the progression of the largest known prime number in ascending order. Here is the Mersenne number with exponent p, where p is a prime number. The longest record-holder was , 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). Durant spent about a year and $2 million on the hunt. This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).

+ Largest known prime by year 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 and D. J. Wheeler 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 NollLandon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer. M23209 6,987 1979 Landon Curt Noll M44497 13,395 1979 David Slowinski and Harry L. Nelson M86243 25,962 1982 David Slowinski M132049 39,751 1983 David Slowinski M216091 65,050 1985 David Slowinski 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.Proof-code: Z, The Prime Pages.Largest non-Mersenne prime that was the largest known prime when it was discovered. M756839 227,832 1992 David Slowinski and Paul Gage M859433 258,716 1994 David Slowinski and Paul Gage M1257787 378,632 1996 David Slowinski and Paul Gage 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, of which the largest 20 are listed below.

Rank Number Discovered Digits Form 1 2136279841 − 1 2024-10-12 41,024,320 Mersenne 2 282589933 − 1 2018-12-07 24,862,048 Mersenne 3 277232917 − 1 2017-12-26 23,249,425 Mersenne 4 274207281 − 1 2016-01-07 22,338,618 Mersenne 5 257885161 − 1 2013-01-25 17,425,170 Mersenne 6 25241902097152 + 1 2025-10-12 13,426,224Generalized Fermat 7 243112609 − 1 2008-08-23 12,978,189 Mersenne 8 242643801 − 1 2009-06-04 12,837,064 Mersenne 9 5166932097152 − 5166931048576 + 1 2023-10-02 11,981,518 Generalized unique 10 4658592097152 − 4658591048576 + 1 2023-05-31 11,887,192 Generalized unique 11 237156667 − 1 2008-09-06 11,185,272 Mersenne 12 232582657 − 1 2006-09-04 9,808,358 Mersenne 13 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth 14 230402457 − 1 2005-12-15 9,152,052 Mersenne 15 4 × 511786358 + 1 2024-10-01 8,238,312 Generalized Proth 16 225964951 − 1 2005-02-18 7,816,230 Mersenne 17 4052186 × 694052186 + 1 2025-04-17 7,451,366 Generalized Cullen 18 69 × 224612729 − 1 2024-08-13 7,409,102 Riesel 19 224036583 − 1 2004-05-15 7,235,733 Mersenne 20 71721621048576 + 1 2026-08-12 7,188,671 Generalized Fermat

See also List of largest known primes and probable primes Integer factorization records

References

External links Press release about the former largest known prime 274,207,281−1 Press release about the former largest known prime 277,232,917−1 Press release about the former largest known prime 282,589,933−1 Press release about the largest known prime 2136,279,841−1