Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2^{p} − 1 for some positive integer *p*. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2^{2} − 1.^{ [1] }^{ [2] } The numbers *p* corresponding to Mersenne primes must themselves be prime, although not all primes *p* lead to Mersenne primes—for example, 2^{11} − 1 = 2047 = 23 × 89.^{ [3] } Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.^{ [2] }^{ [4] }

There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2^{p − 1} × (2^{p} − 1), where 2^{p} − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of *p* = 2, 2^{2} − 1 = 3 is prime, and 2^{2 − 1} × (2^{2} − 1) = 2 × 3 = 6 is perfect.^{ [1] }^{ [5] }^{ [6] }

It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers.^{ [2] }^{ [6] } The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given *x* is (*e*^{γ} / log 2) × log log *x*, where *e* is Euler's number, *γ* is Euler's constant, and log is the natural logarithm.^{ [7] }^{ [8] }^{ [9] } It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a lower bound of 10^{1500}.^{ [10] }

The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents *p*. As of 2022^{ [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 October 2021^{ [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 6 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^{ [lower-alpha 1] } | 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 | c. 1456^{ [lower-alpha 2] } | Anonymous^{ [lower-alpha 3] } | Trial division | ^{ [14] }^{ [15] } |

6 | 17 | 131071 | 6 | 8589869056 | 10 | 1588^{ [lower-alpha 2] } | 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 M. 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^{ [lower-alpha 4] } | 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] } |

* | 59,451,331 | Lowest unverified milestone^{ [lower-alpha 5] } | |||||||

49^{ [lower-alpha 6] } | 74,207,281 | 300376...436351 | 22,338,618 | 451129...315776 | 44,677,235 | January 7, 2016^{ [lower-alpha 7] } | GIMPS / Curtis Cooper | LLT / Prime95 on PC with Intel Core i7-4790 processor | ^{ [61] }^{ [62] } |

50^{ [lower-alpha 6] } | 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^{ [lower-alpha 6] } | 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] } |

* | 107,148,487 | Lowest untested milestone^{ [lower-alpha 5] } |

- ↑ 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. - 1 2 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] } - ↑ Found in an anonymous manuscript,
*Clm 14908*, dated 1456 and 1461^{ [14] }^{ [17] } - ↑
*M*_{42,643,801}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. - 1 2 As of 14 January 2022
^{ [update] }^{ [11] } - 1 2 3 It has not been verified whether any undiscovered Mersenne primes exist between the 48th (
*M*_{57,885,161}) and the 51st (*M*_{82,589,933}) on this table; the ranking is therefore provisional. - ↑
*M*_{74,207,281}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.

**Amicable numbers** are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(*a*)=*b* and σ(*b*)=*a*, where σ(*n*) is equal to the sum of positive divisors of *n*.

The **Great Internet Mersenne Prime Search** (**GIMPS**) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

In mathematics, a **Mersenne prime** is a prime number that is one less than a power of two. That is, it is a prime number of the form *M _{n}* = 2

A **prime number** is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

In number theory, a **perfect number** is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number.

In number theory, a prime number *p* is a **Sophie Germain prime** if 2*p* + 1 is also prime. The number 2*p* + 1 associated with a Sophie Germain prime is called a **safe prime**. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.

In number theory, an odd integer *n* is called an **Euler–Jacobi probable prime** to base *a*, if *a* and *n* are coprime, and

In number theory, a **Woodall number** (*W*_{n}) is any natural number of the form

In number theory, a **Wieferich prime** is a prime number *p* such that *p*^{2} divides 2^{p − 1} − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime *p* divides 2^{p − 1} − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

A **Wilson prime**, named after English mathematician John Wilson, is a prime number *p* such that *p*^{2} divides (*p* − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime *p* divides (*p* − 1)! + 1.

**Prime95**, also distributed as the command-line utility **mprime** for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability.

In mathematics, a **harmonic divisor number**, or **Ore number**, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are:

The number **2,147,483,647** is the eighth Mersenne prime, equal to 2^{31} − 1. It is one of only four known double Mersenne primes.

**Richard Peirce Brent** is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory, random number generators, computer architecture, and analysis of algorithms.

**John Lewis Selfridge**, was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.

The **largest known prime number** is 2^{82,589,933} − 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.

The **SWAC** was an early electronic digital computer built in 1950 by the U.S. National Bureau of Standards (NBS) in Los Angeles, California. It was designed by Harry Huskey.

**Curtis Niles Cooper** is an American mathematician. He currently is a professor at the University of Central Missouri, in the Department of Mathematics and Computer Science.

In mathematics, the **Lucas–Lehmer–Riesel test** is a primality test for numbers of the form *N* = *k* ⋅ 2^{n} − 1, with *k* < 2^{n}. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form *N* = *k* ⋅ 2^{n} + 1, either application of Proth's theorem or one of the deterministic proofs described in Brillhart-Lehmer-Selfridge 1975 are used.

In number theory, **Gillies' conjecture** is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted Lenstra–Pomerance–Wagstaff conjecture.

- 1 2 Stillwell, John (2010).
*Mathematics and Its History*. Undergraduate Texts in Mathematics. Springer Science+Business Media. p. 40. ISBN 978-1-4419-6052-8. Archived from the original on 13 October 2021. Retrieved 13 October 2021. - 1 2 3 4 5 6 7 Caldwell, Chris K. "Mersenne Primes: History, Theorems and Lists".
*PrimePages*. Archived from the original on 4 October 2021. Retrieved 4 October 2021. - ↑ Caldwell, Chris K. "If 2
^{n}-1 is prime, then so is n".*PrimePages*. Archived from the original on 5 October 2021. Retrieved 12 October 2021. - ↑ Prielipp, Robert W. (1970). "Perfect Numbers, Abundant Numbers, and Deficient Numbers".
*The Mathematics Teacher*.**63**(8): 692–96. doi:10.5951/MT.63.8.0692. JSTOR 27958492. Archived from the original on 5 October 2021. Retrieved 13 October 2021– via JSTOR. - ↑ Caldwell, Chris K. "Characterizing all even perfect numbers".
*PrimePages*. Archived from the original on 8 October 2014. Retrieved 12 October 2021. - 1 2 Crilly, Tony (2007). "Perfect numbers".
*50 mathematical ideas you really need to know*. Quercus Publishing. ISBN 978-1-84724-008-8. Archived from the original on 13 October 2021. Retrieved 13 October 2021. - ↑ Caldwell, Chris K. "Heuristics Model for the Distribution of Mersennes".
*PrimePages*. Archived from the original on 5 October 2021. Retrieved 13 October 2021. - ↑ Wagstaff, Samuel S. (January 1983). "Divisors of Mersenne numbers".
*Mathematics of Computation*.**40**(161): 385–397. doi:10.1090/S0025-5718-1983-0679454-X. ISSN 0025-5718. - ↑ Pomerance, Carl (September 1981). "Recent developments in primality testing" (PDF).
*The Mathematical Intelligencer*.**3**(3): 97–105. doi:10.1007/BF03022861. ISSN 0343-6993. S2CID 121750836. - ↑ Ochem, Pascal; Rao, Michaël (30 January 2012). "Odd perfect numbers are greater than 10
^{1500}".*Mathematics of Computation*.**81**(279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. - 1 2 "GIMPS Milestones Report".
*Great Internet Mersenne Prime Search*. Archived from the original on 13 October 2021. Retrieved 7 January 2022. - ↑ Sources applying to almost all entries:
- "List of Known Mersenne Prime Numbers".
*Great Internet Mersenne Prime Search*. Archived from the original on 7 June 2020. Retrieved 4 October 2021. - Caldwell, Chris K. "Mersenne Primes: History, Theorems and Lists".
*PrimePages*. Archived from the original on 4 October 2021. Retrieved 4 October 2021. - Caldwell, Chris K. "The Largest Known prime by Year: A Brief History".
*PrimePages*. Archived from the original on 4 October 2021. Retrieved 13 October 2021. - Haworth, Guy M. (1987). Mersenne numbers (PDF) (Report). Archived (PDF) from the original on 13 October 2021. Retrieved 13 October 2021.
- Noll, Landon Curt (21 December 2018). "Known Mersenne Primes". Archived from the original on 27 July 2021. Retrieved 13 October 2021.
- Tattersall, James J. (1999).
*Elementary Number Theory in Nine Chapters*. Cambridge University Press. pp. 131–134. ISBN 978-0-521-58531-6. Archived from the original on 13 October 2021. Retrieved 13 October 2021.

- "List of Known Mersenne Prime Numbers".
- 1 2 3 4 Joyce, David E. "Euclid's Elements, Book IX, Proposition 36".
*mathcs.clarku.edu*. Archived from the original on 17 June 2021. Retrieved 13 October 2021. - 1 2 3 4 5 6 Dickson, Leonard Eugene (1919).
*History of the Theory of Numbers, Vol. I*. Carnegie Institution of Washington. pp. 4–6. - 1 2 3 4 5 Smith, David Eugene (1925).
*History of Mathematics: Volume II*. Dover. p. 21. ISBN 978-0-486-20430-7. - ↑ O'Connor, John J.; Robertson, Edmund F. "Perfect numbers".
*MacTutor History of Mathematics archive*. Archived from the original on 5 October 2021. Retrieved 13 October 2021. - ↑ "'Calendarium ecclesiasticum – BSB Clm 14908'".
*Bavarian State Library*. Archived from the original on 13 October 2021. Retrieved 13 October 2021. - 1 2 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 October 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. - ↑ Powers, R. E. (November 1911). "The Tenth Perfect Number".
*The American Mathematical Monthly*.**18**(11): 195–197. doi:10.2307/2972574. JSTOR 2972574. - ↑ "Records of Proceedings at Meetings".
*Proceedings of the London Mathematical Society*. s2-13 (1): iv–xl. 1914. doi:10.1112/plms/s2-13.1.1-s. - ↑ 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. - 1 2 "Notes".
*Mathematics of Computation*.**6**(37): 58–61. January 1952. doi:10.1090/S0025-5718-52-99405-2. ISSN 0025-5718. Archived from the original on 13 October 2021. Retrieved 13 October 2021. - ↑ "Notes".
*Mathematics of Computation*.**6**(39): 204–205. July 1952. doi:10.1090/S0025-5718-52-99389-7. ISSN 0025-5718. - 1 2 "Notes".
*Mathematics of Computation*.**7**(41): 67–72. January 1953. doi:10.1090/S0025-5718-53-99372-7. ISSN 0025-5718. - ↑ Riesel, Hans (January 1958). "A New Mersenne Prime".
*Mathematics of Computation*.**12**(61): 60. doi: 10.1090/S0025-5718-58-99282-2 . - 1 2 Hurwitz, Alexander (April 1962). "New Mersenne primes".
*Mathematics of Computation*.**16**(78): 249–251. doi:10.1090/S0025-5718-1962-0146162-X. ISSN 0025-5718. - 1 2 3 Gillies, Donald B. (January 1964). "Three new Mersenne primes and a statistical theory".
*Mathematics of Computation*.**18**(85): 93–97. doi: 10.1090/S0025-5718-1964-0159774-6 . JSTOR 2003409. - ↑ Tuckerman, Bryant (October 1971). "The 24th Mersenne Prime".
*Proceedings of the National Academy of Sciences*.**68**(10): 2319–2320. Bibcode:1971PNAS...68.2319T. doi: 10.1073/pnas.68.10.2319 . PMC 389411 . PMID 16591945. - 1 2 Noll, Landon Curt; Nickel, Laura (October 1980). "The 25th and 26th Mersenne primes".
*Mathematics of Computation*.**35**(152): 1387. doi: 10.1090/S0025-5718-1980-0583517-4 . JSTOR 2006405. - ↑ Slowinski, David (1978). "Searching for the 27th Mersenne prime".
*Journal of Recreational Mathematics*.**11**(4): 258–261. - ↑ "Science Watch: A New Prime Number" .
*The New York Times*. 5 June 1979. Retrieved 13 October 2021. - ↑ "Announcements".
*The Mathematical Intelligencer*.**5**(1): 60. March 1983. doi:10.1007/BF03023507. ISSN 0343-6993. - ↑ Peterson, I. (6 February 1988). "Priming for a Lucky Strike".
*Science News*.**133**(6): 85. doi:10.2307/3972461. JSTOR 3972461. - ↑ Colquitt, W. N.; Welsh, L. (April 1991). "A new Mersenne prime".
*Mathematics of Computation*.**56**(194): 867. Bibcode:1991MaCom..56..867C. doi: 10.1090/S0025-5718-1991-1068823-9 . JSTOR 2008415. - ↑ "Number is largest prime found yet" .
*The Globe and Mail*. 24 September 1983. ProQuest 386439660 – via ProQuest. - ↑ Peterson, I. (28 September 1985). "Prime Time for Supercomputers".
*Science News*.**128**(13): 199. doi:10.2307/3970245. JSTOR 3970245. - ↑ Dembart, Lee (17 September 1985). "Supercomputer Comes Up With Whopping Prime Number".
*Los Angeles Times*. Retrieved 13 October 2021. - ↑ Maddox, John (26 March 1992). "The endless search for primality".
*Nature*.**356**(6367): 283. Bibcode:1992Natur.356..283M. doi:10.1038/356283a0. ISSN 1476-4687. S2CID 4327045. Archived from the original on 13 October 2021. Retrieved 13 October 2021. - ↑ "Largest Known Prime Number Discovered on Cray Research Supercomputer".
*PR Newswire*. 10 January 1994 – via Gale. - ↑ Caldwell, Chris K. "A Prime of Record Size! 2
^{1257787}-1".*PrimePages*. Archived from the original on 5 October 2021. Retrieved 13 October 2021. - ↑ Gillmor, Dan (3 September 1996). "Crunching numbers: Researchers come up with prime math discovery".
*Knight Ridder*– via Gale. - ↑ "GIMPS Discovers 35th Mersenne Prime, 2
^{1,398,269}-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, 2
^{2,976,221}-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, 2
^{3,021,377}-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 2
^{6,972,593}-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, 2
^{13,466,917}-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, 2
^{20,996,011}-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, 2
^{24,036,583}-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. - ↑ "GIMPS Discovers 42nd Mersenne Prime, 2
^{25,964,951}-1 is now the Largest Known Prime".*Great Internet Mersenne Prime Search*. 27 February 2005. Archived from the original on 14 March 2021. Retrieved 13 October 2021. - ↑ "GIMPS Discovers 43rd Mersenne Prime, 2
^{30,402,457}-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, 2
^{32,582,657}-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. - 1 2 "GIMPS Discovers 45th and 46th Mersenne Primes, 2
^{43,112,609}-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. - ↑ Maugh, Thomas H. (27 September 2008). "Rare prime number found".
*Los Angeles Times*. Archived from the original on 27 July 2021. Retrieved 13 October 2021. - ↑ Smith, Edson. "The UCLA Mersenne Prime".
*UCLA Mathematics*. Retrieved 22 November 2021. - ↑ "GIMPS Discovers 48th Mersenne Prime, 2
^{57,885,161}-1 is now the Largest Known Prime".*Great Internet Mersenne Prime Search*. 5 February 2013. Archived from the original on 26 January 2021. Retrieved 13 October 2021. - ↑ Yirka, Bob (6 February 2013). "University professor discovers largest prime number to date".
*phys.org*. Archived from the original on 16 January 2021. Retrieved 13 October 2021. - ↑ "GIMPS Project Discovers Largest Known Prime Number: 2
^{74,207,281}-1".*Great Internet Mersenne Prime Search*. 19 January 2016. Archived from the original on 7 January 2018. Retrieved 13 October 2021. - ↑ "Largest known prime number discovered in Missouri".
*BBC News*. 20 January 2016. Archived from the original on 21 August 2021. Retrieved 13 October 2021. - ↑ "GIMPS Project Discovers Largest Known Prime Number: 2
^{77,232,917}-1".*Great Internet Mersenne Prime Search*. 3 January 2018. Archived from the original on 4 January 2018. Retrieved 13 October 2021. - ↑ Lamb, Evelyn (4 January 2018). "Why You Should Care About a Prime Number That's 23,249,425 Digits Long".
*Slate Magazine*. Archived from the original on 9 October 2021. Retrieved 13 October 2021. - ↑ "GIMPS Discovers Largest Known Prime Number: 2
^{82,589,933}-1".*Great Internet Mersenne Prime Search*. 21 December 2018. Archived from the original on 22 December 2018. Retrieved 13 October 2021. - ↑ Palca, Joe (21 December 2018). "The World Has A New Largest-Known Prime Number".
*NPR*. Archived from the original on 30 July 2021. Retrieved 13 October 2021.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.