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 for some positive integer . For example, is a Mersenne prime as it is a prime number and is expressible as . The exponents corresponding to Mersenne primes must themselves be prime, although the vast majority of primes do not lead to Mersenne primes—for example, .

Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, is a perfect number because the proper divisors of are , and , and .

Euclid proved that every prime expressed as has a corresponding perfect number . For example, the Mersenne prime leads to the corresponding perfect number . 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 is , where is Euler's number, is Euler's constant, and 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 .

The following is a list of all 52 currently known () Mersenne primes and corresponding perfect numbers, along with their exponents . 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 51st () have been tested and verified by GIMPS . Ranks 52 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 Mersenne prime Perfect number Discovery Sources applying to almost all entries:

Value Digits Value Digits Date Discoverer Method 1 2 3 1 6 1 Ancienttimes Known to Ancient Greek mathematicians Unrecorded 2 3 7 1 28 2 3 5 31 2 496 3 4 7 127 3 8128 4 5 13 8191 4 33550336 8 13th centuryor 1456 Ibn Fallus or anonymous Trial division 6 17 131071 6 8589869056 10 Pietro Cataldi 7 19 524287 6 137438691328 12 8 31 2147483647 10 230584...952128 19 Leonhard Euler Trial division with modular restrictions 9 61 230584...693951 19 265845...842176 37 Ivan Pervushin Lucas sequences 10 89 618970...562111 27 191561...169216 54 Ralph Ernest Powers 11 107 162259...288127 33 131640...728128 65 12 127 170141...105727 39 144740...152128 77 Édouard Lucas 13 521 686479...057151 157 235627...646976 314 Raphael M. Robinson Lucas–Lehmer test (LLT) on SWAC 14 607 531137...728127 183 141053...328128 366 15 1,279 104079...729087 386 541625...291328 770 16 2,203 147597...771007 664 108925...782528 1,327 17 2,281 446087...836351 687 994970...915776 1,373 18 3,217 259117...315071 969 335708...525056 1,937 Hans Riesel LLT on BESK 19 4,253 190797...484991 1,281 182017...377536 2,561 Alexander Hurwitz LLT on IBM 7090 20 4,423 285542...580607 1,332 407672...534528 2,663 21 9,689 478220...754111 2,917 114347...577216 5,834 Donald B. Gillies LLT on ILLIAC II 22 9,941 346088...463551 2,993 598885...496576 5,985 23 11,213 281411...392191 3,376 395961...086336 6,751 24 19,937 431542...041471 6,002 931144...942656 12,003 Bryant Tuckerman LLT on IBM 360/91 25 21,701 448679...882751 6,533 100656...605376 13,066 Landon Curt Noll & Laura Nickel LLT on CDC Cyber 174 26 23,209 402874...264511 6,987 811537...666816 13,973 Landon Curt Noll 27 44,497 854509...228671 13,395 365093...827456 26,790 Harry L. Nelson & David Slowinski LLT on Cray-1 28 86,243 536927...438207 25,962 144145...406528 51,924 David Slowinski 29 110,503 521928...515007 33,265 136204...862528 66,530 Walter Colquitt & Luke Welsh LLT on NEC SX-2 30 132,049 512740...061311 39,751 131451...550016 79,502 David Slowinski et al. (Cray) LLT on Cray X-MP 31 216,091 746093...528447 65,050 278327...880128 130,100 LLT on Cray X-MP/24 32 756,839 174135...677887 227,832 151616...731328 455,663 LLT on Harwell Lab's Cray-2 33 859,433 129498...142591 258,716 838488...167936 517,430 LLT on Cray C90 34 1,257,787 412245...366527 378,632 849732...704128 757,263 LLT on Cray T94 35 1,398,269 814717...315711 420,921 331882...375616 841,842 GIMPS / Joel Armengaud LLT / Prime95 on 90 MHz Pentium PC 36 2,976,221 623340...201151 895,932 194276...462976 1,791,864 GIMPS / Gordon Spence LLT / Prime95 on 100 MHz Pentium PC 37 3,021,377 127411...694271 909,526 811686...457856 1,819,050 GIMPS / Roland Clarkson LLT / Prime95 on 200 MHz Pentium PC 38 6,972,593 437075...193791 2,098,960 955176...572736 4,197,919 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 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 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 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 GIMPS / Martin Nowak 43 30,402,457 315416...943871 9,152,052 497437...704256 18,304,103 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 45 37,156,667 202254...220927 11,185,272 204534...480128 22,370,543 GIMPS / Hans-Michael Elvenich LLT / Prime95 on PC 46 42,643,801 169873...314751 12,837,064 144285...253376 25,674,127 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 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 GIMPS / Curtis Cooper LLT / Prime95 on PC at University of Central Missouri 49 74,207,281 300376...436351 22,338,618 451129...315776 44,677,235 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 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 GIMPS / Patrick Laroche LLT / Prime95 on PC with Intel Core i5-4590T processor * 83,200,483 Lowest unverified milestone 52 136,279,841 881694...871551 41,024,320 388692...008576 82,048,640 GIMPS / Luke Durant LLT / PRPLL on Nvidia H100 GPU * 141,924,943 Lowest untested milestone

Notes

References

External links

List on GIMPS, with the full values of large numbers A technical report on the history of Mersenne numbers, by Guy Haworth