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 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[update], 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
editThe 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[update], the previously discovered prime M82589933, having 24,862,048 digits, held the record for more than 6 years, longer than any other prime since M19937 (which held the record for 7 years from 1971 to 1978).[citation needed]
Prizes
editThere 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
editThis section needs additional citations for verification. (October 2024) |
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, |
44 | 1951 | Aimé Ferrier with a mechanical calculator; the largest record not set by computer. | |
| 180×(M127)2+1 | 521064401567922879406069432539 |
79 | 1951 | J. C. P. Miller & D. J. Wheeler[14] Using Cambridge's EDSAC computer |
| M521 | 686479766013060971498190079908 |
157 | 1952 | Raphael M. Robinson |
| M607 | 531137992816767098689588206552 |
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
editA 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
editReferences
edit- ^ 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.
- ^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
- ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
- ^ 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.
- ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019.
An interesting side note is about the binary representations of those numbers...
- ^ McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.
- ^ Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.
- ^ "List of known Mersenne prime numbers - PrimeNet". www.mersenne.org. "41024320" link is to a zip file with the digits. Retrieved 2024-10-21.
- ^ 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.
- ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
- ^ "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.
- ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
- ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. The Mathematical Association of America. ISBN 9780883855843.
- ^ Miller, J. C. P. (1951). "Large Prime Numbers". Nature. 168 (4280): 838. Bibcode:1951Natur.168..838M. doi:10.1038/168838b0.
- ^ a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
- ^ 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.
- ^ Proof-code: Z, The Prime Pages.
- ^ "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
- ^ "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
- ^ "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.
- ^ 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.
- ^ "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.
- ^ "PrimePage Primes: Phi(3, - 516693^1048576)". t5k.org.
- ^ "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
- ^ "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.
- ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
- ^ "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.
- ^ "4 × 511786358 + 1". t5k.org. PrimePages. 1 October 2024. Retrieved 5 October 2024.
- ^ "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.
- ^ "69 × 224612729 − 1". t5k.org. PrimePages. 13 August 2024. Retrieved 29 August 2024.
- ^ "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.
- ^ "107347 × 223427517 − 1". t5k.org. PrimePages. 4 August 2024. Retrieved 25 August 2024.
- ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com.[dead link]
- ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.