# List of Mersenne primes and perfect numbers

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. The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2$^{11}$ − 1 = 2047 = 23 × 89.
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.
Euclid proved c. 300 BCE that every prime expressed as M$_p$ = 2$^p$ − 1 has a corresponding perfect number M$_p$ × (M$_p$+1)/2 = 2$^{p−1}$ × (2$^p$ − 1). For example, the Mersenne prime 2$^2$ − 1 = 3 leads to the corresponding perfect number 2$^{2−1}$ × (2$^2$ − 1) = 2 × 3 = 6. In 1747, Leonhard Euler completed what is now called the Euclid–Euler theorem, showing that these are the only even perfect numbers. As a result, there is a one-to-one correspondence between Mersenne primes and even perfect numbers, so a list of one can be converted into a list of the other.
It is currently an open problem whether there are infinitely many Mersenne primes and even perfect numbers. The density 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. It is widely believed, but not proven, that no odd perfect numbers exist; numerous restrictive conditions have been proven, including a lower bound of 10$^{1500}$.
The following is a list of all 52 currently known (as of January 2025) Mersenne primes and corresponding perfect numbers, along with their exponents p. The largest 18 of these have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS; their discoverers are listed as "GIMPS / name", where the name is the person who supplied the computer that made the discovery. New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. Due to this efficiency, the largest known prime number has often been a Mersenne prime.
All possible exponents up to the 48th (p = 57,885,161) have been tested and verified by GIMPS as of October 2021. Ranks 49 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Later entries are extremely long, so only the first and last six digits of each number are shown, along with the number of decimal digits.
Table of all 52 currently-known Mersenne primes and corresponding perfect numbers :
| Rank | p | Mersenne prime | Mersenne prime | Perfect number | Perfect number | Discovery | Discovery | Discovery | Ref. |
| Rank | p | Value | Digits | Value | Digits | Date | Discoverer | Method | Ref. |
| 1 | 2 | 3 | 1 | 6 | 1 | Ancient times | Known to Ancient Greek mathematicians | Unrecorded | |
| 2 | 3 | 7 | 1 | 28 | 2 | Ancient times | Known to Ancient Greek mathematicians | Unrecorded | |
| 3 | 5 | 31 | 2 | 496 | 3 | Ancient times | Known to Ancient Greek mathematicians | Unrecorded | |
| 4 | 7 | 127 | 3 | 8128 | 4 | Ancient times | Known to Ancient Greek mathematicians | Unrecorded | |
| 5 | 13 | 8191 | 4 | 33550336 | 8 | 13th century or 1456 | Ibn Fallus or anonymous | Trial division | |
| 6 | 17 | 131071 | 6 | 8589869056 | 10 | 1588 | Pietro Cataldi | Trial division | |
| 7 | 19 | 524287 | 6 | 137438691328 | 12 | 1588 | Pietro Cataldi | Trial division | |
| 8 | 31 | 2147483647 | 10 | 230584...952128 | 19 | 1772 | Leonhard Euler | Trial division with modular restrictions | |
| 9 | 61 | 230584...693951 | 19 | 265845...842176 | 37 | Nov 1883 | Ivan Pervushin | Lucas sequences | |
| 10 | 89 | 618970...562111 | 27 | 191561...169216 | 54 | Jun 1911 | Ralph Ernest Powers | Lucas sequences | |
| 11 | 107 | 162259...288127 | 33 | 131640...728128 | 65 | Jun 1, 1914 | Ralph Ernest Powers | Lucas sequences | |
| 12 | 127 | 170141...105727 | 39 | 144740...152128 | 77 | Jan 10, 1876 | Édouard Lucas | Lucas sequences | |
| 13 | 521 | 686479...057151 | 157 | 235627...646976 | 314 | Jan 30, 1952 | Raphael M. Robinson | Lucas–Lehmer test (LLT) on SWAC | |
| 14 | 607 | 531137...728127 | 183 | 141053...328128 | 366 | Jan 30, 1952 | Raphael M. Robinson | Lucas–Lehmer test (LLT) on SWAC | |
| 15 | 1,279 | 104079...729087 | 386 | 541625...291328 | 770 | Jun 25, 1952 | Raphael M. Robinson | Lucas–Lehmer test (LLT) on SWAC | |
| 16 | 2,203 | 147597...771007 | 664 | 108925...782528 | 1,327 | Oct 7, 1952 | Raphael M. Robinson | Lucas–Lehmer test (LLT) on SWAC | |
| 17 | 2,281 | 446087...836351 | 687 | 994970...915776 | 1,373 | Oct 9, 1952 | Raphael M. Robinson | Lucas–Lehmer test (LLT) on SWAC | |
| 18 | 3,217 | 259117...315071 | 969 | 335708...525056 | 1,937 | Sep 8, 1957 | Hans Riesel | LLT on BESK | |
| 19 | 4,253 | 190797...484991 | 1,281 | 182017...377536 | 2,561 | Nov 3, 1961 | Alexander Hurwitz | LLT on IBM 7090 | |
| 20 | 4,423 | 285542...580607 | 1,332 | 407672...534528 | 2,663 | Nov 3, 1961 | Alexander Hurwitz | LLT on IBM 7090 | |
| 21 | 9,689 | 478220...754111 | 2,917 | 114347...577216 | 5,834 | May 11, 1963 | Donald B. Gillies | LLT on ILLIAC II | |
| 22 | 9,941 | 346088...463551 | 2,993 | 598885...496576 | 5,985 | May 16, 1963 | Donald B. Gillies | LLT on ILLIAC II | |
| 23 | 11,213 | 281411...392191 | 3,376 | 395961...086336 | 6,751 | Jun 2, 1963 | Donald B. Gillies | LLT on ILLIAC II | |
| 24 | 19,937 | 431542...041471 | 6,002 | 931144...942656 | 12,003 | Mar 4, 1971 | Bryant Tuckerman | LLT on IBM 360/91 | |
| 25 | 21,701 | 448679...882751 | 6,533 | 100656...605376 | 13,066 | Oct 30, 1978 | Landon Curt Noll & Laura Nickel | LLT on CDC Cyber 174 | |
| 26 | 23,209 | 402874...264511 | 6,987 | 811537...666816 | 13,973 | Feb 9, 1979 | Landon Curt Noll | LLT on CDC Cyber 174 | |
| 27 | 44,497 | 854509...228671 | 13,395 | 365093...827456 | 26,790 | Apr 8, 1979 | Harry L. Nelson & David Slowinski | LLT on Cray-1 | |
| 28 | 86,243 | 536927...438207 | 25,962 | 144145...406528 | 51,924 | Sep 25, 1982 | David Slowinski | LLT on Cray-1 | |
| 29 | 110,503 | 521928...515007 | 33,265 | 136204...862528 | 66,530 | Jan 29, 1988 | Walter Colquitt & Luke Welsh | LLT on NEC SX-2 | |
| 30 | 132,049 | 512740...061311 | 39,751 | 131451...550016 | 79,502 | Sep 19, 1983 | David Slowinski et al. (Cray) | LLT on Cray X-MP | |
| 31 | 216,091 | 746093...528447 | 65,050 | 278327...880128 | 130,100 | Sep 1, 1985 | David Slowinski et al. (Cray) | LLT on Cray X-MP/24 | |
| 32 | 756,839 | 174135...677887 | 227,832 | 151616...731328 | 455,663 | Feb 17, 1992 | David Slowinski et al. (Cray) | LLT on Harwell Lab's Cray-2 | |
| 33 | 859,433 | 129498...142591 | 258,716 | 838488...167936 | 517,430 | Jan 4, 1994 | David Slowinski et al. (Cray) | LLT on Cray C90 | |
| 34 | 1,257,787 | 412245...366527 | 378,632 | 849732...704128 | 757,263 | Sep 3, 1996 | David Slowinski et al. (Cray) | LLT on Cray T94 | |
| 35 | 1,398,269 | 814717...315711 | 420,921 | 331882...375616 | 841,842 | Nov 13, 1996 | GIMPS / Joel Armengaud | LLT / Prime95 on 90 MHz Pentium PC | |
| 36 | 2,976,221 | 623340...201151 | 895,932 | 194276...462976 | 1,791,864 | Aug 24, 1997 | GIMPS / Gordon Spence | LLT / Prime95 on 100 MHz Pentium PC | |
| 37 | 3,021,377 | 127411...694271 | 909,526 | 811686...457856 | 1,819,050 | Jan 27, 1998 | GIMPS / Roland Clarkson | LLT / Prime95 on 200 MHz Pentium PC | |
| 38 | 6,972,593 | 437075...193791 | 2,098,960 | 955176...572736 | 4,197,919 | Jun 1, 1999 | GIMPS / Nayan Hajratwala | LLT / Prime95 on IBM Aptiva with 350 MHz Pentium II processor | |
| 39 | 13,466,917 | 924947...259071 | 4,053,946 | 427764...021056 | 8,107,892 | Nov 14, 2001 | GIMPS / Michael Cameron | LLT / Prime95 on PC with 800 MHz Athlon T-Bird processor | |
| 40 | 20,996,011 | 125976...682047 | 6,320,430 | 793508...896128 | 12,640,858 | Nov 17, 2003 | GIMPS / Michael Shafer | LLT / Prime95 on Dell Dimension PC with 2 GHz Pentium 4 processor | |
| 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 | |
| 42 | 25,964,951 | 122164...077247 | 7,816,230 | 746209...088128 | 15,632,458 | Feb 18, 2005 | GIMPS / Martin Nowak | LLT / Prime95 on PC with 2.4 GHz Pentium 4 processor | |
| 43 | 30,402,457 | 315416...943871 | 9,152,052 | 497437...704256 | 18,304,103 | Dec 15, 2005 | GIMPS / Curtis Cooper & Steven Boone | LLT / Prime95 on PC at University of Central Missouri | |
| 44 | 32,582,657 | 124575...967871 | 9,808,358 | 775946...120256 | 19,616,714 | Sep 4, 2006 | GIMPS / Curtis Cooper & Steven Boone | LLT / Prime95 on PC at University of Central Missouri | |
| 45 | 37,156,667 | 202254...220927 | 11,185,272 | 204534...480128 | 22,370,543 | Sep 6, 2008 | GIMPS / Hans-Michael Elvenich | LLT / Prime95 on PC | |
| 46 | 42,643,801 | 169873...314751 | 12,837,064 | 144285...253376 | 25,674,127 | Jun 4, 2009 | GIMPS / Odd Magnar Strindmo | LLT / Prime95 on PC with 3 GHz Intel Core 2 processor | |
| 47 | 43,112,609 | 316470...152511 | 12,978,189 | 500767...378816 | 25,956,377 | Aug 23, 2008 | GIMPS / Edson Smith | LLT / Prime95 on Dell OptiPlex PC with Intel Core 2 Duo E6600 processor | |
| 48 | 57,885,161 | 581887...285951 | 17,425,170 | 169296...130176 | 34,850,340 | Jan 25, 2013 | GIMPS / Curtis Cooper | LLT / Prime95 on PC at University of Central Missouri | |
| * | 72,072,997 | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone | Lowest unverified milestone |
| 49 | 74,207,281 | 300376...436351 | 22,338,618 | 451129...315776 | 44,677,235 | Jan 7, 2016 | GIMPS / Curtis Cooper | LLT / Prime95 on PC with Intel Core i7-4790 processor | |
| 50 | 77,232,917 | 467333...179071 | 23,249,425 | 109200...301056 | 46,498,850 | Dec 26, 2017 | GIMPS / Jonathan Pace | LLT / Prime95 on PC with Intel Core i5-6600 processor | |
| 51 | 82,589,933 | 148894...902591 | 24,862,048 | 110847...207936 | 49,724,095 | Dec 7, 2018 | GIMPS / Patrick Laroche | LLT / Prime95 on PC with Intel Core i5-4590T processor | |
| * | 130,439,863 | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone | Lowest untested milestone |
| 52 | 136,279,841 | 881694...871551 | 41,024,320 | 388692...008576 | 82,048,640 | Oct 12, 2024 | GIMPS / Luke Durant | LLT / PRPLL on Nvidia H100 GPU | |