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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 to the Great Internet Mersenne Prime Search (GIMPS).[1]

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. According to Euclid's theorem there are infinitely many prime numbers, so 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, the seven largest known primes are Mersenne primes.[2] The last eighteen record primes were Mersenne primes.[3][4] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.[5]

Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not the case.[6][7]

Current record

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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:[8]

881694327503833265553939100378117358971207354509066041067156376412422630694756841441725990347723283108837509739959776874 ...

(41,024,080 digits skipped)

... 852806517931459412567957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

As of October 2024, the previously discovered prime M82589933, having 24,862,048 digits, held the record for almost 6 years, longer than any other prime since M19937 (which held the record for 7 years from 1971 to 1978).[citation needed]

Prizes

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There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[9] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[10] In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[9] Time called this prime the 29th top invention of 2008.[11]

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.[9]

GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[12]

History

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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.[3] 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. No records are known prior to 1456.[citation needed]

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 Luke Durant using a network of thousands of dedicated graphics processing units.[1]

Number Decimal expansion
(partial for numbers > M1000) 		Digits Year found Discoverer
M13 8,191 4 1456 Anonymous
M17 131,071 6 1588 Pietro Cataldi
M19 524,287 6 1588 Pietro Cataldi
6,700,417 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.[13]
M31 2,147,483,647 10 1772 Leonhard Euler
999,999,000,001 12 1851 Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record.
67,280,421,310,721 14 1855 Thomas Clausen (but no proof was provided).
M127 170,141,183,460,469,
231,731,687,303,715,
884,105,727 		39 1876 Édouard Lucas
20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 44 1951 Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1 5210644015679228794060694325390955853335898483908056458352183851018372555735221 79 1951 J. C. P. Miller & D. J. Wheeler[14]
Using Cambridge's EDSAC computer
M521 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151 157 1952 Raphael M. Robinson
M607 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127 183 1952 Raphael M. Robinson
M1279 104079321946...703168729087 386 1952 Raphael M. Robinson
M2203 147597991521...686697771007 664 1952 Raphael M. Robinson
M2281 446087557183...418132836351 687 1952 Raphael M. Robinson
M3217 259117086013...362909315071 969 1957 Hans Riesel
M4423 285542542228...902608580607 1,332 1961 Alexander Hurwitz
M9689 478220278805...826225754111 2,917 1963 Donald B. Gillies
M9941 346088282490...883789463551 2,993 1963 Donald B. Gillies
M11213 281411201369...087696392191 3,376 1963 Donald B. Gillies
M19937 431542479738...030968041471 6,002 1971 Bryant Tuckerman
M21701 448679166119...353511882751 6,533 1978 Laura A. Nickel and Landon Curt Noll[15]
M23209 402874115778...523779264511 6,987 1979 Landon Curt Noll[15]
M44497 854509824303...961011228671 13,395 1979 David Slowinski and Harry L. Nelson[15]
M86243 536927995502...709433438207 25,962 1982 David Slowinski[15]
M132049 512740276269...455730061311 39,751 1983 David Slowinski[15]
M216091 746093103064...103815528447 65,050 1985 David Slowinski[15]
148140632376...836387377151 65,087 1989 A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[16][17]
Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839 174135906820...328544677887 227,832 1992 David Slowinski and Paul Gage[15]
M859433 129498125604...243500142591 258,716 1994 David Slowinski and Paul Gage[15]
M1257787 412245773621...976089366527 378,632 1996 David Slowinski and Paul Gage[15]
M1398269 814717564412...868451315711 420,921 1996 GIMPS, Joel Armengaud
M2976221 623340076248...743729201151 895,932 1997 GIMPS, Gordon Spence
M3021377 127411683030...973024694271 909,526 1998 GIMPS, Roland Clarkson
M6972593 437075744127...142924193791 2,098,960 1999 GIMPS, Nayan Hajratwala
M13466917 924947738006...470256259071 4,053,946 2001 GIMPS, Michael Cameron
M20996011 125976895450...762855682047 6,320,430 2003 GIMPS, Michael Shafer
M24036583 299410429404...882733969407 7,235,733 2004 GIMPS, Josh Findley
M25964951 122164630061...280577077247 7,816,230 2005 GIMPS, Martin Nowak
M30402457 315416475618...411652943871 9,152,052 2005 GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M32582657 124575026015...154053967871 9,808,358 2006 GIMPS, Curtis Cooper and Steven Boone
M43112609 316470269330...166697152511 12,978,189 2008 GIMPS, Edson Smith
M57885161 581887266232...071724285951 17,425,170 2013 GIMPS, Curtis Cooper
M74207281 300376418084...391086436351 22,338,618 2016 GIMPS, Curtis Cooper
M77232917 467333183359...069762179071 23,249,425 2017 GIMPS, Jonathan Pace
M82589933 148894445742...325217902591 24,862,048 2018 GIMPS, Patrick Laroche
M136279841 881694327503...219486871551 41,024,320 2024 GIMPS, Luke Durant

Twenty largest

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A list of the 5,000 largest known primes is maintained by the PrimePages,[18] of which the twenty largest are listed below.[19]

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 [20]
3 277232917 − 1 2017-12-26 23,249,425 Mersenne [21]
4 274207281 − 1 2016-01-07 22,338,618 Mersenne [22]
5 257885161 − 1 2013-01-25 17,425,170 Mersenne [23]
6 243112609 − 1 2008-08-23 12,978,189 Mersenne [24]
7 242643801 − 1 2009-06-04 12,837,064 Mersenne [25]
8 Φ3(−5166931048576) 2023-10-02 11,981,518 Generalized unique [26]
9 Φ3(−4658591048576) 2023-05-31 11,887,192 Generalized unique [27]
10 237156667 − 1 2008-09-06 11,185,272 Mersenne [24]
11 232582657 − 1 2006-09-04 9,808,358 Mersenne [28]
12 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [29]
13 230402457 − 1 2005-12-15 9,152,052 Mersenne [30]
14 4 × 511786358 + 1 2024-10-01 8,238,312 Generalized Proth [31]
15 225964951 − 1 2005-02-18 7,816,230 Mersenne [32]
16 69 × 224612729 − 1 2024-08-13 7,409,102 Riesel [33]
17 224036583 − 1 2004-05-15 7,235,733 Mersenne [34]
18 107347 × 223427517 − 1 2024-08-04 7,052,391 Riesel [35]
19 3 × 222103376 − 1 2024-09-30 6,653,780 Thabit [36]
20 19637361048576 + 1 2022-09-24 6,598,776 Generalized Fermat [37]

See also

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References

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  1. ^ a b c d "GIMPS Project Discovers Largest Known Prime Number: 2136,279,841-1". Mersenne Research, Inc. 21 October 2024. Retrieved 21 October 2024.
  2. ^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
  3. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
  4. ^ 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.
  5. ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
  6. ^ McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.
  7. ^ Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.
  8. ^ "List of known Mersenne prime numbers - PrimeNet". www.mersenne.org. "41024320" link is to a zip file with the digits. Retrieved 2024-10-21.
  9. ^ a b c "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
  10. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
  11. ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
  12. ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
  13. ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. The Mathematical Association of America. ISBN 9780883855843.
  14. ^ Miller, J. C. P. (1951). "Large Prime Numbers". Nature. 168 (4280): 838. Bibcode:1951Natur.168..838M. doi:10.1038/168838b0.
  15. ^ a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
  16. ^ Brown, John; Noll, Landon Curt; Parady, B. K.; Smith, Joel F.; Zarantonello, Sergio E.; Smith, Gene Ward; Robinson, Raphael M.; Andrews, George E. (1990). "Letters to the Editor". The American Mathematical Monthly. 97 (3): 214–215. doi:10.1080/00029890.1990.11995576. JSTOR 2324686.
  17. ^ Proof-code: Z, The Prime Pages.
  18. ^ "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
  19. ^ "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
  20. ^ "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
  21. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
  22. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
  23. ^ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
  24. ^ a b "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
  25. ^ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
  26. ^ "PrimePage Primes: Phi(3, - 516693^1048576)". t5k.org.
  27. ^ "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
  28. ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
  29. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
  30. ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
  31. ^ "4 × 511786358 + 1". t5k.org. PrimePages. 1 October 2024. Retrieved 5 October 2024.
  32. ^ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
  33. ^ "69 × 224612729 − 1". t5k.org. PrimePages. 13 August 2024. Retrieved 29 August 2024.
  34. ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
  35. ^ "107347 × 223427517 − 1". t5k.org. PrimePages. 4 August 2024. Retrieved 25 August 2024.
  36. ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com.[dead link‍]
  37. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.
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