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Largest known prime number

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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).

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. , 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 2k − 1 is simply k ones.

Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not the case. Primes with millions of digits are not useful for cryptography.

Current record

The record is currently held by 2136,279,841 − 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 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was , which was the largest known prime for 184 years.

The primes up to and including \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). Durant spent about one year and US$2 million on the hunt. This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).

NumberDigitsYear foundDiscoverer
M1761588Pietro Cataldi
M1961588Pietro Cataldi
M31101772Leonhard Euler
\mathsf{\tfrac{M_{59}}{179951}}131867Fortuné Landry
M127391876Édouard Lucas
\mathsf{\tfrac{2^{148}+1}{17}}441951Aimé Ferrier, with a mechanical calculator. The largest record not set by computer.
180×(M127)2+1791951J. C. P. Miller & D. J. Wheeler using Cambridge's EDSAC computer
M5211571952Raphael M. Robinson
M6071831952Raphael M. Robinson
M12793861952Raphael M. Robinson
M22036641952Raphael M. Robinson
M22816871952Raphael M. Robinson
M32179691957Hans Riesel
M44231,3321961Alexander Hurwitz
M96892,9171963Donald B. Gillies
M99412,9931963Donald B. Gillies
M112133,3761963Donald B. Gillies
M199376,0021971Bryant Tuckerman
M217016,5331978Laura A. Nickel and Landon Curt Noll
M232096,9871979Landon Curt Noll
M4449713,3951979David Slowinski and Harry L. Nelson
M8624325,9621982David Slowinski
M13204939,7511983David Slowinski
M21609165,0501985David Slowinski
391581×2216193−165,0871989The "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.
M756839227,8321992David Slowinski and Paul Gage
M859433258,7161994David Slowinski and Paul Gage
M1257787378,6321996David Slowinski and Paul Gage
M1398269420,9211996GIMPS, Joel Armengaud
M2976221895,9321997GIMPS, Gordon Spence
M3021377909,5261998GIMPS, Roland Clarkson
M69725932,098,9601999GIMPS, Nayan Hajratwala
M134669174,053,9462001GIMPS, Michael Cameron
M209960116,320,4302003GIMPS, Michael Shafer
M240365837,235,7332004GIMPS, Josh Findley
M259649517,816,2302005GIMPS, Martin Nowak
M304024579,152,0522005GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M325826579,808,3582006GIMPS, Curtis Cooper and Steven Boone
M4311260912,978,1892008GIMPS, Edson Smith
M5788516117,425,1702013GIMPS, Curtis Cooper
M7420728122,338,6182016GIMPS, Curtis Cooper
M7723291723,249,4252017GIMPS, Jonathan Pace
M8258993324,862,0482018GIMPS, Patrick Laroche
M13627984141,024,3202024GIMPS, 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.

RankNumberDiscoveredDigitsFormRef
12136279841 − 12024-10-1241,024,320Mersenne
2282589933 − 12018-12-0724,862,048Mersennetitle=GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1url=https://www.mersenne.org/primes/press/M82589933.htmldate=21 December 2018work=Mersenne Research, Inc.access-date=21 December 2018 }}
3277232917 − 12017-12-2623,249,425Mersennetitle=GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1url=https://www.mersenne.org/primes/press/M77232917.htmlwebsite=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=3 January 2018}}
4274207281 − 12016-01-0722,338,618Mersennetitle=GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1url=https://www.mersenne.org/primes/?press=M74207281website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017}}
5257885161 − 12013-01-2517,425,170Mersennetitle=GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M57885161website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=5 February 2013}}
625241902097152 + 12025-10-1213,426,224Generalized Fermat
7243112609 − 12008-08-2312,978,189Mersennetitle=GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M43112609website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=15 September 2008}}
8242643801 − 12009-06-0412,837,064Mersennetitle=GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime.url=https://www.mersenne.org/primes/?press=M42643801website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=12 April 2009}}
9Φ3(−5166931048576)2023-10-0211,981,518Generalized unique
10Φ3(−4658591048576)2023-05-3111,887,192Generalized unique
11237156667 − 12008-09-0611,185,272Mersenne
12232582657 − 12006-09-049,808,358Mersennetitle=GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M32582657website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=11 September 2006}}
1310223 × 231172165 + 12016-10-319,383,761Prothtitle=PrimeGrid's Seventeen or Bust Subprojecturl=http://www.primegrid.com/download/SOB-31172165.pdfwebsite=primegrid.compublisher=PrimeGridaccess-date=30 September 2017}}
14230402457 − 12005-12-159,152,052Mersennetitle=GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M30402457website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=24 December 2005}}
154 × 511786358 + 12024-10-018,238,312Generalized Proth
16225964951 − 12005-02-187,816,230Mersennetitle=GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M25964951website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=27 February 2005}}
174052186 × 694052186 + 12025-04-177,451,366Generalized Cullen
1869 × 224612729 − 12024-08-137,409,102Riesel
19224036583 − 12004-05-157,235,733Mersennetitle=GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime.url=https://www.mersenne.org/primes/?press=M24036583website=mersenne.orgpublisher=Great Internet Mersenne Prime Searchaccess-date=29 September 2017date=28 May 2004}}
2053362841048576 + 12025-11-027,054,022Generalized Fermat

References

References

  1. (21 October 2024). "GIMPS Project Discovers Largest Known Prime Number: 2136,279,841-1". Mersenne Research, Inc..
  2. "A 41-million-digit prime number is the biggest ever found—but mathematicians' search for perfection will continue".
  3. "The largest known primes – Database Search Output". Prime Pages.
  4. "The Largest Known Prime by Year: A Brief History". Prime Pages.
  5. The last non-Mersenne to be the largest known prime, was [http://t5k.org/primes/page.php?id=390 391,581 ⋅ 2216,193 − 1]; see also [http://t5k.org/notes/by_year.html The Largest Known Prime by year: A Brief History] originally by Caldwell.
  6. "Perfect Numbers".
  7. Johnston, Nathaniel. (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography".
  8. (January 22, 2016). "PSA: Do Not Use The New Prime Number For RSA Encryption".
  9. Great_Internet_Mersenne_Prime_Search#endnote_number_size^_‡M
  10. "List of known Mersenne prime numbers - PrimeNet".
  11. Electronic Frontier Foundation, [https://www.eff.org/press/releases/big-prime-nets-big-prize Big Prime Nets Big Prize].
  12. (October 14, 2009). "Record 12-Million-Digit Prime Number Nets $100,000 Prize". [[Electronic Frontier Foundation]].
  13. (October 29, 2008). "Best Inventions of 2008 - 29. The 46th Mersenne Prime". [[Time Inc]].
  14. "GIMPS by Mersenne Research, Inc.".
  15. Brasch, Ben. (October 23, 2024). "One year, 41 million digits: How he found the largest known prime number". [[Washington Post]].
  16. Bragg, Julianna. (2024-11-01). "World's largest known prime number found by former Nvidia programmer".
  17. McRae, Mike. (2024-10-25). "Amateur Discovers The Largest Known Prime Number And It's Huge".
  18. Miller, J. C. P.. (1951). "Large Prime Numbers". Nature.
  19. [[Landon Curt Noll]], [http://www.isthe.com/chongo/tech/math/prime/prime_press.html Large Prime Number Found by SGI/Cray Supercomputer].
  20. (1990). "Letters to the Editor". The American Mathematical Monthly.
  21. [https://t5k.org/bios/code.php?code=Z Proof-code: Z], The [[Prime Pages]].
  22. "The Prime Database: The List of Largest Known Primes Home Page".
  23. "The Top Twenty: Largest Known Primes".
  24. (21 December 2018). "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc..
  25. "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". [[Great Internet Mersenne Prime Search]].
  26. "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". [[Great Internet Mersenne Prime Search]].
  27. (5 February 2013). "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  28. "PrimeGrid's Generalized Fermat Prime Search". [[PrimeGrid]].
  29. (15 September 2008). "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  30. (12 April 2009). "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime.". [[Great Internet Mersenne Prime Search]].
  31. "PrimePage Primes: Phi(3, - 516693^1048576)".
  32. "PrimePage Primes: Phi(3, - 465859^1048576)".
  33. (11 September 2006). "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  34. "PrimeGrid's Seventeen or Bust Subproject". [[PrimeGrid]].
  35. (24 December 2005). "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  36. (1 October 2024). "4 × 511786358 + 1". [[PrimePages]].
  37. (27 February 2005). "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  38. "PrimePage Primes: 4052186 ×69^4052186 + 1".
  39. (13 August 2024). "69 × 224612729 − 1". [[PrimePages]].
  40. (28 May 2004). "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime.". [[Great Internet Mersenne Prime Search]].
  41. (2 November 2024). "53362841048576 + 1". [[PrimePages]].
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