# Largest known prime number Information

https://en.wikipedia.org/wiki/Largest_known_prime_number

The largest known prime number (as of November 2022) 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 largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

A prime number is a positive integer, excluding 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 utilise a specialised primality test that is faster than the general one. As of November 2022, the eight 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 1's, since the binary form of 2k − 1 is simply k 1's. [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]

## Prizes

The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3,000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits. [7] There are several prizes offered by the Electronic Frontier Foundation for record primes. [8] GIMPS is also coordinating its long-range search efforts for primes of over 100 million digits and larger and will split the Electronic Frontier Foundation's US$150,000 prize with a winning participant.

The record passed one million digits in 1999, earning a US$50,000 prize. [9] In 2008, the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation. [8] Time called this prime the 29th top invention of 2008. [10] Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS. Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits. [8]

## 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
${\displaystyle {\tfrac {2^{32}+1}{641}}}$ 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
${\displaystyle {\tfrac {10^{18}+1}{1000001}}}$ 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.
${\displaystyle {\tfrac {2^{64}+1}{274177}}}$ 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
${\displaystyle {\tfrac {2^{148}+1}{17}}}$ 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
M607

531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127

183 1952
M1279 104079321946...703168729087 386 1952
M2203 147597991521...686697771007 664 1952
M2281 446087557183...418132836351 687 1952
M3217 259117086013...362909315071 969 1957 Hans Riesel
M4423 285542542228...902608580607 1,332 1961
M9689 478220278805...826225754111 2,917 1963
M9941 346088282490...883789463551 2,993 1963
M11213 281411201369...087696392191 3,376 1963
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]
${\displaystyle 391581\times 2^{216193}-1}$ 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 Chris K. Caldwell, [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 2 43112609 − 1 2008-08-23 12,978,189 Mersenne [21]
6 242643801 − 1 2009-06-04 12,837,064 Mersenne [22]
7 237156667 − 1 2008-09-06 11,185,272 Mersenne [21]
8 232582657 − 1 2006-09-04 9,808,358 Mersenne [23]
9 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [24]
10 230402457 − 1 2005-12-15 9,152,052 Mersenne [25]
11 225964951 − 1 2005-02-18 7,816,230 Mersenne [26]
12 224036583 − 1 2004-05-15 7,235,733 Mersenne [27]
13 19637361048576 + 1 2022-09-24 6,598,776 Generalized Fermat [28]
14 19517341048576 + 1 2022-08-09 6,595,985 Generalized Fermat [29]
15 202705 × 221320516 + 1 2021-12-01 6,418,121 Proth [30]
16 220996011 − 1 2003-11-17 6,320,430 Mersenne [31]
17 10590941048576 + 1 2018-10-31 6,317,602 Generalized Fermat [32]
18 9194441048576 + 1 2017-08-29 6,253,210 Generalized Fermat [33]
19 7 × 220267500 + 1 2022-07-21 6,101,127 Proth [34]
20 168451 × 219375200 + 1 2017-09-17 5,832,522 Proth [35]

## References

1. ^ a b c "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
2. ^ Caldwell, Chris. "The largest known primes – Database Search Output". Prime Pages. Retrieved June 3, 2018.
3. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016.
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 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. ^
7. ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
8. ^ 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.
9. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
10. ^ "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.
11. ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. ISBN  9780883855843.
12. ^ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
13. ^ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
14. ^ Proof-code: Z, The Prime Pages.
15. ^ "The Prime Database: The List of Largest Known Primes Home Page". primes.utm.edu/primes. Chris K. Caldwell. Retrieved 30 September 2017.
16. ^ "The Top Twenty: Largest Known Primes". Chris K. Caldwell. Retrieved 3 January 2018.
17. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
18. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
19. ^ "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.
20. ^ 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.
21. ^ "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.
22. ^ "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.
23. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
24. ^ "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.
25. ^ "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.
26. ^ "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.
27. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.
28. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 17 September 2022.
29. ^ "PrimeGrid's Extended Sierpinski Problem Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 28 December 2021.
30. ^ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.
31. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
32. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
33. ^ "PrimePage Primes: 7 x 2^20267500 + 1". Retrieved 25 July 2022.
34. ^ "PrimeGrid's Prime Sierpinski Problem" (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.