List of Mersenne primes and perfect numbers
There is a
It is currently an
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2023[update], there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS.[2] New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.[2]
The displayed ranks are among indices currently known as of 2022[update]; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of January 2024[update].[11] The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last six digits of each number are shown.
Rank | p | Mersenne prime | Mersenne prime digits | Perfect number | Perfect number digits | Discovery | Discoverer | Method | Ref.[12] |
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 1 | 6 | 1 | Ancient times[a] | Known to Ancient Greek mathematicians
|
Unrecorded | [13][14][15] |
2 | 3 | 7 | 1 | 28 | 2 | [13][14][15] | |||
3 | 5 | 31 | 2 | 496 | 3 | [13][14][15] | |||
4 | 7 | 127 | 3 | 8128
|
4 | [13][14][15] | |||
5 | 13 | 8191 | 4 | 33550336 | 8 | 1200s/c. 1456[b] | Multiple[c] | Trial division | [14][15] |
6 | 17 | 131071 | 6 | 8589869056 | 10 | 1588[b] | Pietro Cataldi | [2][18] | |
7 | 19 | 524287 | 6 | 137438691328 | 12 | [2][18] | |||
8 | 31 | 2147483647 | 10 | 230584...952128 | 19 | 1772 | Leonhard Euler | Trial division with modular restrictions | [19][20] |
9 | 61 | 230584...693951 | 19 | 265845...842176 | 37 | November 1883 | Ivan Pervushin | Lucas sequences | [21] |
10 | 89 | 618970...562111 | 27 | 191561...169216 | 54 | June 1911 | Ralph Ernest Powers | [22] | |
11 | 107 | 162259...288127 | 33 | 131640...728128 | 65 | June 1, 1914 | [23] | ||
12 | 127 | 170141...105727 | 39 | 144740...152128 | 77 | January 10, 1876 | Édouard Lucas | [24] | |
13 | 521 | 686479...057151 | 157 | 235627...646976 | 314 | January 30, 1952 | Raphael M. Robinson | LLT on SWAC | [25] |
14 | 607 | 531137...728127 | 183 | 141053...328128 | 366 | [25] | |||
15 | 1,279 | 104079...729087 | 386 | 541625...291328 | 770 | June 25, 1952 | [26] | ||
16 | 2,203 | 147597...771007 | 664 | 108925...782528 | 1,327 | October 7, 1952 | [27] | ||
17 | 2,281 | 446087...836351 | 687 | 994970...915776 | 1,373 | October 9, 1952 | [27] | ||
18 | 3,217 | 259117...315071 | 969 | 335708...525056 | 1,937 | September 8, 1957 | Hans Riesel | LLT on BESK | [28] |
19 | 4,253 | 190797...484991 | 1,281 | 182017...377536 | 2,561 | November 3, 1961 | Alexander Hurwitz | LLT on IBM 7090 | [29] |
20 | 4,423 | 285542...580607 | 1,332 | 407672...534528 | 2,663 | [29] | |||
21 | 9,689 | 478220...754111 | 2,917 | 114347...577216 | 5,834 | May 11, 1963 | Donald B. Gillies | LLT on ILLIAC II | [30] |
22 | 9,941 | 346088...463551 | 2,993 | 598885...496576 | 5,985 | May 16, 1963 | [30] | ||
23 | 11,213 | 281411...392191 | 3,376 | 395961...086336 | 6,751 | June 2, 1963 | [30] | ||
24 | 19,937 | 431542...041471 | 6,002 | 931144...942656 | 12,003 | March 4, 1971 | Bryant Tuckerman | LLT on IBM 360 /91
|
[31] |
25 | 21,701 | 448679...882751 | 6,533 | 100656...605376 | 13,066 | October 30, 1978 | Landon Curt Noll & Laura Nickel | LLT on CDC Cyber 174 | [32] |
26 | 23,209 | 402874...264511 | 6,987 | 811537...666816 | 13,973 | February 9, 1979 | Landon Curt Noll | [32] | |
27 | 44,497 | 854509...228671 | 13,395 | 365093...827456 | 26,790 | April 8, 1979 | Harry L. Nelson & David Slowinski | LLT on Cray-1 | [33][34] |
28 | 86,243 | 536927...438207 | 25,962 | 144145...406528 | 51,924 | September 25, 1982 | David Slowinski | [35] | |
29 | 110,503 | 521928...515007 | 33,265 | 136204...862528 | 66,530 | January 29, 1988 | Walter Colquitt & Luke Welsh | LLT on NEC SX-2 | [36][37] |
30 | 132,049 | 512740...061311 | 39,751 | 131451...550016 | 79,502 | September 19, 1983 | David Slowinski et al. (Cray) | LLT on Cray X-MP | [38] |
31 | 216,091 | 746093...528447 | 65,050 | 278327...880128 | 130,100 | September 1, 1985 | LLT on Cray X-MP/24 | [39][40] | |
32 | 756,839 | 174135...677887 | 227,832 | 151616...731328 | 455,663 | February 17, 1992 | LLT on Harwell Lab's Cray-2 | [41] | |
33 | 859,433 | 129498...142591 | 258,716 | 838488...167936 | 517,430 | January 4, 1994 | LLT on Cray C90 | [42] | |
34 | 1,257,787 | 412245...366527 | 378,632 | 849732...704128 | 757,263 | September 3, 1996 | LLT on Cray T94 | [43][44] | |
35 | 1,398,269 | 814717...315711 | 420,921 | 331882...375616 | 841,842 | November 13, 1996 | GIMPS / Joel Armengaud | LLT / Prime95 on 90 MHz Pentium PC | [45] |
36 | 2,976,221 | 623340...201151 | 895,932 | 194276...462976 | 1,791,864 | August 24, 1997 | GIMPS / Gordon Spence | LLT / Prime95 on 100 MHz Pentium PC | [46] |
37 | 3,021,377 | 127411...694271 | 909,526 | 811686...457856 | 1,819,050 | January 27, 1998 | GIMPS / Roland Clarkson | LLT / Prime95 on 200 MHz Pentium PC | [47] |
38 | 6,972,593 | 437075...193791 | 2,098,960 | 955176...572736 | 4,197,919 | June 1, 1999 | GIMPS / Nayan Hajratwala | LLT / Prime95 on IBM Aptiva with 350 MHz Pentium II processor | [48] |
39 | 13,466,917 | 924947...259071 | 4,053,946 | 427764...021056 | 8,107,892 | November 14, 2001 | GIMPS / Michael Cameron | LLT / Prime95 on PC with 800 MHz Athlon T-Bird processor | [49] |
40 | 20,996,011 | 125976...682047 | 6,320,430 | 793508...896128 | 12,640,858 | November 17, 2003 | GIMPS / Michael Shafer | LLT / Prime95 on Dell Dimension PC with 2 GHz Pentium 4 processor | [50] |
41 | 24,036,583 | 299410...969407 | 7,235,733 | 448233...950528 | 14,471,465 | May 15, 2004 | GIMPS / Josh Findley | LLT / Prime95 on PC with 2.4 GHz Pentium 4 processor | [51] |
42 | 25,964,951 | 122164...077247 | 7,816,230 | 746209...088128 | 15,632,458 | February 18, 2005 | GIMPS / Martin Nowak | [52] | |
43 | 30,402,457 | 315416...943871 | 9,152,052 | 497437...704256 | 18,304,103 | December 15, 2005 | GIMPS / Curtis Cooper & Steven Boone | LLT / Prime95 on PC at University of Central Missouri | [53] |
44 | 32,582,657 | 124575...967871 | 9,808,358 | 775946...120256 | 19,616,714 | September 4, 2006 | [54] | ||
45 | 37,156,667 | 202254...220927 | 11,185,272 | 204534...480128 | 22,370,543 | September 6, 2008 | GIMPS / Hans-Michael Elvenich | LLT / Prime95 on PC | [55] |
46 | 42,643,801 | 169873...314751 | 12,837,064 | 144285...253376 | 25,674,127 | June 4, 2009[d] | GIMPS / Odd Magnar Strindmo | LLT / Prime95 on PC with 3 GHz Intel Core 2 processor | [56] |
47 | 43,112,609 | 316470...152511 | 12,978,189 | 500767...378816 | 25,956,377 | August 23, 2008 | GIMPS / Edson Smith | LLT / Prime95 on Dell OptiPlex PC with Intel Core 2 Duo E6600 processor | [55][57][58] |
48 | 57,885,161 | 581887...285951 | 17,425,170 | 169296...130176 | 34,850,340 | January 25, 2013 | GIMPS / Curtis Cooper | LLT / Prime95 on PC at University of Central Missouri | [59][60] |
* | 67,242,060 | Lowest unverified milestone[e] | |||||||
49[f] | 74,207,281 | 300376...436351 | 22,338,618 | 451129...315776 | 44,677,235 | January 7, 2016[g] | GIMPS / Curtis Cooper | LLT / Prime95 on PC with Intel Core i7-4790 processor | [61][62] |
50[f] | 77,232,917 | 467333...179071 | 23,249,425 | 109200...301056 | 46,498,850 | December 26, 2017 | GIMPS / Jonathan Pace | LLT / Prime95 on PC with Intel Core i5-6600 processor | [63][64] |
51[f] | 82,589,933 | 148894...902591 | 24,862,048 | 110847...207936 | 49,724,095 | December 7, 2018 | GIMPS / Patrick Laroche | LLT / Prime95 on PC with Intel Core i5-4590T processor | [65][66] |
* | 115,388,888 | Lowest untested milestone[e] |
Historically, the largest known prime number has often been a Mersenne prime.
Notes
- ^ The first four perfect numbers were documented by Nicomachus circa 100, and the concept was known (along with corresponding Mersenne primes) to Euclid at the time of his Elements. There is no record of discovery.
- ^ Islamic mathematicians such as Ismail ibn Ibrahim ibn Fallus (1194–1239) may have known of the fifth through seventh perfect numbers prior to European records.[16]
- Clm 14908, dated 1456 and 1461, and in Ibn Fallus' earlier work, which was not widely distributed[14][17]
- ^ M42,643,801 was first reported to GIMPS on April 12, 2009 but was not noticed by a human until June 4, 2009 due to a server error.
- ^ a b As of 10 March 2024[update][11]
- ^ a b c It has not been verified whether any undiscovered Mersenne primes exist between the 48th (M57,885,161) and the 51st (M82,589,933) on this table; the ranking is therefore provisional.
- ^ M74,207,281 was first reported to GIMPS on September 17, 2015 but was not noticed by a human until January 7, 2016 due to a server error.
References
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- ^ Caldwell, Chris K. "Characterizing all even perfect numbers". PrimePages. Archived from the original on 8 October 2014. Retrieved 12 October 2021.
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- ^ a b c d e f Dickson, Leonard Eugene (1919). History of the Theory of Numbers, Vol. I. Carnegie Institution of Washington. pp. 4–6.
- ^ ISBN 978-0-486-20430-7.
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- ^ "'Calendarium ecclesiasticum – BSB Clm 14908'". Bavarian State Library. Archived from the original on 13 October 2021. Retrieved 13 October 2021.
- ^ a b Cataldi, Pietro Antonio (1603). Trattato de' numeri perfetti di Pietro Antonio Cataldo [Pietro Antonio Cataldi's treatise on perfect numbers] (in Italian). Presso di Heredi di Giouanni Rossi.
- ^ Caldwell, Chris K. "Modular restrictions on Mersenne divisors". PrimePages. Retrieved 22 November 2021.
- ^ Euler, Leonhard (1772). "Extrait d'un lettre de M. Euler le pere à M. Bernoulli concernant le Mémoire imprimé parmi ceux de 1771, p 318" [Extract of a letter from Mr. Euler to Mr. Bernoulli, concerning the Mémoire published among those of 1771]. Nouveaux Mémoires de l'académie royale des sciences de Berlin (in French). 1772: 35–36. Archived from the original on 15 August 2020. Retrieved 13 October 2021 – via Euler Archive.
- ^ "Sur un nouveau nombre premier, annoncé par le père Pervouchine" [On a new prime number, announced by Pervouchine]. Bulletin de l'Académie impériale des sciences de St.-Pétersbourg (in French). 31: 532–533. 27 January 1887. Archived from the original on 13 October 2021. Retrieved 13 October 2021 – via Biodiversity Heritage Library.
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- ^ Lucas, Édouard (1876). "Note sur l'application des séries récurrentes à la recherche de la loi de distribution des nombres premiers" [Note on the application of recurrent series to researching the law of prime number distribution]. Comptes rendus de l'Académie des Sciences (in French). 82: 165–167. Archived from the original on 13 October 2021. Retrieved 13 October 2021.
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- ^ Gillmor, Dan (3 September 1996). "Crunching numbers: Researchers come up with prime math discovery". Knight Ridder – via Gale.
- ^ "GIMPS Discovers 35th Mersenne Prime, 21,398,269-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 12 November 1996. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 36th Mersenne Prime, 22,976,221-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 1 September 1997. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 37th Mersenne Prime, 23,021,377-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 2 February 1998. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 38th Mersenne Prime 26,972,593-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 30 June 1999. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 39th Mersenne Prime, 213,466,917-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 6 December 2001. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 2 February 2003. Archived from the original on 7 June 2020. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 28 May 2004. Archived from the original on 29 January 2021. Retrieved 13 October 2021.
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- ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 24 December 2005. Archived from the original on 14 March 2021. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 11 September 2006. Archived from the original on 26 January 2021. Retrieved 13 October 2021.
- ^ a b "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. Archived from the original on 5 October 2021. Retrieved 13 October 2021.
- ^ "GIMPS Discovers 47th Mersenne Prime". Great Internet Mersenne Prime Search. 12 April 2009. Archived from the original on 19 February 2021. Retrieved 13 October 2021.
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