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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).[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. 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.[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]

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 100 digits of its value are:[9]

8816943275038332655539391003781173589712073545090660410671563764124226306947568414417259903477232831 ...

(41,024,120 digits skipped)

... 7957568284228288124096109707961148305849349766085764170715060409404509622104665555076706219486871551

Prizes

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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

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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.[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.

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]

Primes found without a computer[4]
Number Decimal expansion Digits Year found Discoverer
M17 131071 6 1588 Pietro Cataldi
M19 524287 6 1588 Pietro Cataldi
M31 2147483647 10 1772 Leonhard Euler
3203431780337 13 1867 Fortuné Landry [fr]
M127 17014118346046923173168
7303715884105727 		39 1876 Édouard Lucas
20988936657440586486151
264256610222593863921 		44 1951 Aimé Ferrier, with a mechanical calculator. The largest record not set by computer.
Primes found using computers[4]
Number Digits Year found Discoverer
180×(M127)2+1 79 1951 J. C. P. Miller & D. J. Wheeler[17] 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[18]
M23209 6,987 1979 Landon Curt Noll[18]
M44497 13,395 1979 David Slowinski and Harry L. Nelson[18]
M86243 25,962 1982 David Slowinski[18]
M132049 39,751 1983 David Slowinski[18]
M216091 65,050 1985 David Slowinski[18]
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.[19][20]
Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839 227,832 1992 David Slowinski and Paul Gage[18]
M859433 258,716 1994 David Slowinski and Paul Gage[18]
M1257787 378,632 1996 David Slowinski and Paul Gage[18]
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

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A list of the 5,000 largest known primes is maintained by the PrimePages,[21] of which the twenty largest are listed below.[22]

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 [23]
3 277232917 − 1 2017-12-26 23,249,425 Mersenne [24]
4 274207281 − 1 2016-01-07 22,338,618 Mersenne [25]
5 257885161 − 1 2013-01-25 17,425,170 Mersenne [26]
6 243112609 − 1 2008-08-23 12,978,189 Mersenne [27]
7 242643801 − 1 2009-06-04 12,837,064 Mersenne [28]
8 Φ3(−5166931048576) 2023-10-02 11,981,518 Generalized unique [29]
9 Φ3(−4658591048576) 2023-05-31 11,887,192 Generalized unique [30]
10 237156667 − 1 2008-09-06 11,185,272 Mersenne [27]
11 232582657 − 1 2006-09-04 9,808,358 Mersenne [31]
12 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [32]
13 230402457 − 1 2005-12-15 9,152,052 Mersenne [33]
14 4 × 511786358 + 1 2024-10-01 8,238,312 Generalized Fermat [34]
15 225964951 − 1 2005-02-18 7,816,230 Mersenne [35]
16 69 × 224612729 − 1 2024-08-13 7,409,102 Riesel [36]
17 224036583 − 1 2004-05-15 7,235,733 Mersenne [37]
18 107347 × 223427517 − 1 2024-08-04 7,052,391 Riesel [38]
19 38432361048576 + 1 2024-12-22 6,904,556 Generalized Fermat [39]
20 3 × 222103376 − 1 2024-09-30 6,653,780 Thabit [40]

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. ^ Voight, John; Conversation, The. "A 41-million-digit prime number is the biggest ever found—but mathematicians' search for perfection will continue". phys.org. Retrieved 2025-01-14.
  3. ^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
  4. ^ a b c d Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
  5. ^ 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.
  6. ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
  7. ^ McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.
  8. ^ Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.
  9. ^ "List of known Mersenne prime numbers - PrimeNet". www.mersenne.org. "41024320" link is to a zip file with the digits. Retrieved 2024-10-21.
  10. ^ 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.
  11. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
  12. ^ "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.
  13. ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
  14. ^ Brasch, Ben (October 23, 2024). "One year, 41 million digits: How he found the largest known prime number". Washington Post. Retrieved April 9, 2025.
  15. ^ Bragg, Julianna (2024-11-01). "World's largest known prime number found by former Nvidia programmer". CNN. Retrieved 2024-11-28.
  16. ^ McRae, Mike (2024-10-25). "Amateur Discovers The Largest Known Prime Number And It's Huge". ScienceAlert. Retrieved 2024-11-28.
  17. ^ Miller, J. C. P. (1951). "Large Prime Numbers". Nature. 168 (4280): 838. Bibcode:1951Natur.168..838M. doi:10.1038/168838b0.
  18. ^ a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
  19. ^ 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.
  20. ^ Proof-code: Z, The Prime Pages.
  21. ^ "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
  22. ^ "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
  23. ^ "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
  24. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
  25. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
  26. ^ "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.
  27. ^ 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.
  28. ^ "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.
  29. ^ "PrimePage Primes: Phi(3, - 516693^1048576)". t5k.org.
  30. ^ "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
  31. ^ "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.
  32. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
  33. ^ "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.
  34. ^ "4 × 511786358 + 1". t5k.org. PrimePages. 1 October 2024. Retrieved 5 October 2024.
  35. ^ "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.
  36. ^ "69 × 224612729 − 1". t5k.org. PrimePages. 13 August 2024. Retrieved 29 August 2024.
  37. ^ "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.
  38. ^ "107347 × 223427517 − 1". t5k.org. PrimePages. 4 August 2024. Retrieved 25 August 2024.
  39. ^ "PrimePage Primes: 3843236^1048576 + 1". t5k.org. Retrieved 2025-04-09.
  40. ^ "PrimePage Primes: 3 · 2^22103376 - 1". t5k.org. Retrieved 2025-04-09.
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