Largest known prime number

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The largest known prime number ((As of December 2023)) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.[1]

A 2020 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 June 2023), the six largest known primes are Mersenne primes.[2] The last seventeen 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]

Current record

The record is currently held by 282,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018.[1] The first and last 120 digits of its value are shown below:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...

(24,861,808 digits skipped)

... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]

This prime has held the record for 5 years (as of December 2023), longer than any other record prime since M19937 (which held the record for 7 years, 1971–1978).

Prizes

There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[7] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[8] In 2008, a ten-million digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[7] Time (magazine) called this prime the 29th top invention of 2008.[9]

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 prize is offered for the first prime with at least one billion digits.[7]

GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[10]

History of largest known prime numbers

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. 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.

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
[math]\displaystyle{ \tfrac{2^{32}+1}{641} }[/math] 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.[11]
M31 2,147,483,647 10 1772 Leonhard Euler
[math]\displaystyle{ \tfrac{10^{18}+1}{1000001} }[/math] 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.
[math]\displaystyle{ \tfrac{2^{64}+1}{274177} }[/math] 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
[math]\displaystyle{ \tfrac{2^{148}+1}{17} }[/math] 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[12]
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[13]
M23209 402874115778...523779264511 6,987 1979 Landon Curt Noll[13]
M44497 854509824303...961011228671 13,395 1979 David Slowinski and Harry L. Nelson[13]
M86243 536927995502...709433438207 25,962 1982 David Slowinski[13]
M132049 512740276269...455730061311 39,751 1983 David Slowinski[13]
M216091 746093103064...103815528447 65,050 1985 David Slowinski[13]
[math]\displaystyle{ 391581 \times 2^{216193} - 1 }[/math] 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.[14][15]
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[13]
M859433 129498125604...243500142591 258,716 1994 David Slowinski and Paul Gage[13]
M1257787 412245773621...976089366527 378,632 1996 David Slowinski and Paul Gage[13]
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

GIMPS found the fifteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.

The twenty largest known prime numbers

A list of the 5,000 largest known primes is maintained by the PrimePages,[16] of which the twenty largest are listed below.[17]

Rank Number Discovered Digits Form Ref
1 282589933 − 1 2018-12-07 24,862,048 Mersenne [1]
2 277232917 − 1 2017-12-26 23,249,425 Mersenne [18]
3 274207281 − 1 2016-01-07 22,338,618 Mersenne [19]
4 257885161 − 1 2013-01-25 17,425,170 Mersenne [20]
5 243112609 − 1 2008-08-23 12,978,189 Mersenne [21]
6 242643801 − 1 2009-06-04 12,837,064 Mersenne [22]
7 Φ3(−5166931048576) 2023-10-02 11,981,518 Generalized unique [23]
8 Φ3(−4658591048576) 2023-05-31 11,887,192 Generalized unique [24]
9 237156667 − 1 2008-09-06 11,185,272 Mersenne [21]
10 232582657 − 1 2006-09-04 9,808,358 Mersenne [25]
11 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [26]
12 230402457 − 1 2005-12-15 9,152,052 Mersenne [27]
13 225964951 − 1 2005-02-18 7,816,230 Mersenne [28]
14 224036583 − 1 2004-05-15 7,235,733 Mersenne [29]
15 19637361048576 + 1 2022-09-24 6,598,776 Generalized Fermat [30]
16 19517341048576 + 1 2022-08-09 6,595,985 Generalized Fermat [31]
17 202705 × 221320516 + 1 2021-12-01 6,418,121 Proth [32]
18 220996011 − 1 2003-11-17 6,320,430 Mersenne [33]
19 10590941048576 + 1 2018-10-31 6,317,602 Generalized Fermat [34]
20 3 × 220928756 − 1 2023-07-05 6,300,184 Thabit [35]

See also

  • List of largest known primes and probable primes

References

  1. ↑ 1.0 1.1 1.2 "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc.. 21 December 2018. https://www.mersenne.org/primes/press/M82589933.html. 
  2. ↑ "The largest known primes – Database Search Output". Prime Pages. https://t5k.org/primes/search.php?Number=100. 
  3. ↑ 3.0 3.1 Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. http://t5k.org/notes/by_year.html. 
  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". http://www.personal.psu.edu/sxt104/class/Math140H/PerfectNum.html. "An interesting side note is about the binary representations of those numbers..." 
  6. ↑ "51st Known Mersenne Prime Discovered". https://www.mersenne.org/primes/press/M82589933.html. 
  7. ↑ 7.0 7.1 7.2 "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. https://www.eff.org/press/archives/2009/10/14-0. 
  8. ↑ Electronic Frontier Foundation, Big Prime Nets Big Prize.
  9. ↑ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time (Time Inc). October 29, 2008. http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html. 
  10. ↑ "GIMPS by Mersenne Research, Inc.". https://www.mersenne.org/legal/. 
  11. ↑ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. The Mathematical Association of America. ISBN 9780883855843. https://books.google.com/books?id=3c6iBQAAQBAJ&pg=PA43. 
  12. ↑ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
  13. ↑ 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
  14. ↑ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
  15. ↑ Proof-code: Z, The Prime Pages.
  16. ↑ "The Prime Database: The List of Largest Known Primes Home Page". https://t5k.org/primes/home.php. 
  17. ↑ "The Top Twenty: Largest Known Primes". https://t5k.org/top20/page.php?id=3. 
  18. ↑ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". Great Internet Mersenne Prime Search. https://www.mersenne.org/primes/press/M77232917.html. 
  19. ↑ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". Great Internet Mersenne Prime Search. https://www.mersenne.org/primes/?press=M74207281. 
  20. ↑ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 5 February 2013. https://www.mersenne.org/primes/?press=M57885161. 
  21. ↑ 21.0 21.1 "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 15 September 2008. https://www.mersenne.org/primes/?press=M43112609. 
  22. ↑ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime.". Great Internet Mersenne Prime Search. 12 April 2009. https://www.mersenne.org/primes/?press=M42643801. 
  23. ↑ "PrimePage Primes: Phi(3, - 516693^1048576)". https://t5k.org/primes/page.php?id=136490. 
  24. ↑ "PrimePage Primes: Phi(3, - 465859^1048576)". https://t5k.org/primes/page.php?id=136107. 
  25. ↑ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 11 September 2006. https://www.mersenne.org/primes/?press=M32582657. 
  26. ↑ "PrimeGrid's Seventeen or Bust Subproject". PrimeGrid. http://www.primegrid.com/download/SOB-31172165.pdf. 
  27. ↑ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 24 December 2005. https://www.mersenne.org/primes/?press=M30402457. 
  28. ↑ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 27 February 2005. https://www.mersenne.org/primes/?press=M25964951. 
  29. ↑ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 28 May 2004. https://www.mersenne.org/primes/?press=M24036583. 
  30. ↑ "PrimeGrid's Generalized Fermat Prime Search". PrimeGrid. https://www.primegrid.com/download/GFN-1963736_1048576.pdf. 
  31. ↑ "PrimeGrid's Generalized Fermat Prime Search". PrimeGrid. https://www.primegrid.com/download/GFN-1951734_1048576.pdf. 
  32. ↑ "PrimeGrid's Extended Sierpinski Problem Prime Search". PrimeGrid. http://www.primegrid.com/download/ESP-202705.pdf. 
  33. ↑ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. 2 December 2003. https://www.mersenne.org/primes/?press=M20996011. 
  34. ↑ "PrimeGrid's Generalized Fermat Prime Search". PrimeGrid. https://www.primegrid.com/download/GFN-1059094_1048576.pdf. 
  35. ↑ "PrimeGrid's 321 Prime Search". PrimeGrid. https://www.primegrid.com/download/321-20928756.pdf. 

External links



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