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^{[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 | c. 1456^{[b]} | Anonymous^{[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 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^{[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]} |
* | 59,451,331 | 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]} |
* | 107,148,487 | Lowest untested milestone^{[e]} |
NotesEdit
- ^ 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.
- ^ ^{a} ^{b} 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.
- ^ ^{a} ^{b} As of 14 January 2022^{[update]}^{[11]}
- ^ ^{a} ^{b} ^{c} 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.
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