# List of perfect numbers

The following is a list of the known perfect numbers, along with the Mersenne prime exponent *p* which generates them with the expression 2^{p−1}× (2^{p} − 1) where 2^{p} − 1 is a Mersenne prime. All even perfect numbers are of this form, it is not known whether there are any odd perfect numbers.^{[1]} As of 2018^{[update]} there are 50 known perfect numbers in total.^{[2]}^{[3]}^{[4]} The ratio *p* / digits approaches log(10) / log(4) = 1.6609640474...

Rank | p |
Perfect number | Digits | Year | Discoverer |
---|---|---|---|---|---|

1 | 2 | 6 | 1 | 4th century B.C.^{[5]} |
Euclid |

2 | 3 | 28 | 2 | 4th century B.C. | Euclid |

3 | 5 | 496 | 3 | 4th century B.C. | Euclid |

4 | 7 | 8128 | 4 | 4th century B.C. | Euclid |

5 | 13 | 33550336 | 8 | 1456 | First seen in a medieval manuscript, Munich, Bayerische Staatsbibliothek, CLM 14908, fol. 33^{[6]} |

6 | 17 | 8589869056 | 10 | 1588 | Cataldi^{[1]} |

7 | 19 | 137438691328 | 12 | 1588 | Cataldi^{[1]} |

8 | 31 | 2305843008139952128 | 19 | 1772 | Euler |

9 | 61 | 265845599156...615953842176 | 37 | 1883 | Pervushin |

10 | 89 | 191561942608...321548169216 | 54 | 1911 | Powers |

11 | 107 | 131640364585...117783728128 | 65 | 1914 | Powers |

12 | 127 | 144740111546...131199152128 | 77 | 1876 | Lucas |

13 | 521 | 235627234572...160555646976 | 314 | 1952 | Robinson |

14 | 607 | 141053783706...759537328128 | 366 | 1952 | Robinson |

15 | 1,279 | 541625262843...764984291328 | 770 | 1952 | Robinson |

16 | 2,203 | 108925835505...834453782528 | 1,327 | 1952 | Robinson |

17 | 2,281 | 994970543370...675139915776 | 1,373 | 1952 | Robinson |

18 | 3,217 | 335708321319...332628525056 | 1,937 | 1957 | Riesel |

19 | 4,253 | 182017490401...437133377536 | 2,561 | 1961 | Hurwitz |

20 | 4,423 | 407672717110...642912534528 | 2,663 | 1961 | Hurwitz |

21 | 9,689 | 114347317530...558429577216 | 5,834 | 1963 | Gillies |

22 | 9,941 | 598885496387...324073496576 | 5,985 | 1963 | Gillies |

23 | 11,213 | 395961321281...702691086336 | 6,751 | 1963 | Gillies |

24 | 19,937 | 931144559095...790271942656 | 12,003 | 1971 | Tuckerman |

25 | 21,701 | 100656497054...255141605376 | 13,066 | 1978 | Noll & Nickel |

26 | 23,209 | 811537765823...603941666816 | 13,973 | 1979 | Noll |

27 | 44,497 | 365093519915...353031827456 | 26,790 | 1979 | Nelson & Slowinski |

28 | 86,243 | 144145836177...957360406528 | 51,924 | 1982 | Slowinski |

29 | 110,503 | 136204582133...233603862528 | 66,530 | 1988 | Colquitt & Welsh |

30 | 132,049 | 131451295454...491774550016 | 79,502 | 1983 | Slowinski |

31 | 216,091 | 278327459220...416840880128 | 130,100 | 1985 | Slowinski |

32 | 756,839 | 151616570220...600565731328 | 455,663 | 1992 | Slowinski & Gage |

33 | 859,433 | 838488226750...540416167936 | 517,430 | 1994 | Slowinski & Gage |

34 | 1,257,787 | 849732889343...028118704128 | 757,263 | 1996 | Slowinski & Gage |

35 | 1,398,269 | 331882354881...017723375616 | 841,842 | 1996 | Armengaud, Woltman, et al. |

36 | 2,976,221 | 194276425328...724174462976 | 1,791,864 | 1997 | Spence, Woltman, et al. |

37 | 3,021,377 | 811686848628...573022457856 | 1,819,050 | 1998 | Clarkson, Woltman, Kurowski, et al. |

38 | 6,972,593 | 955176030521...475123572736 | 4,197,919 | 1999 | Hajratwala, Woltman, Kurowski, et al. |

39 | 13,466,917 | 427764159021...460863021056 | 8,107,892 | 2001 | Cameron, Woltman, Kurowski, et al. |

40 | 20,996,011 | 793508909365...578206896128 | 12,640,858 | 2003 | Shafer, Woltman, Kurowski, et al. |

41 | 24,036,583 | 448233026179...460572950528 | 14,471,465 | 2004 | Findley, Woltman, Kurowski, et al. |

42 | 25,964,951 | 746209841900...874791088128 | 15,632,458 | 2005 | Nowak, Woltman, Kurowski, et al. |

43 | 30,402,457 | 497437765459...536164704256 | 18,304,103 | 2005 | Cooper, Boone, Woltman, Kurowski, et al. |

44 | 32,582,657 | 775946855336...476577120256 | 19,616,714 | 2006 | Cooper, Boone, Woltman, Kurowski, et al. |

45 | 37,156,667 | 204534225534...975074480128 | 22,370,543 | 2008 | Elvenich, Woltman, Kurowski, et al. |

46 | 42,643,801 | 144285057960...837377253376 | 25,674,127 | 2009 | Strindmo, Woltman, Kurowski, et al. |

47 | 43,112,609 | 500767156849...221145378816 | 25,956,377 | 2008 | Smith, Woltman, Kurowski, et al. |

48 | 57,885,161 | 169296395301...626270130176 | 34,850,340 | 2013 | Cooper, Woltman, Kurowski, et al. |

49 | 74,207,281 | 451129962706...557930315776 | 44,677,235 | 2016 | Cooper, Woltman, Kurowski, Blosser, et al. |

50 | 77,232,917 | 109200152134...402016301056 | 46,498,850 | 2017 | Pace, Woltman, Kurowski, Blosser, et al. |

The displayed ranks are among those perfect numbers which are known as of April 2018^{[update]}. Some ranks may change later if smaller perfect numbers are discovered, it is known there is no odd perfect number below 10^{1500}.^{[7]} GIMPS reported that by 8 April 2018 the search for Mersenne primes (and thereby even perfect numbers) became exhaustive up to the 47th above.^{[8]}

## References[edit]

- ^
^{a}^{b}^{c}Crilly, Tony (2007).*50 mathematical ideas you really need to know*. Quercus Publishing. p. 43. ISBN 978-1-84724-008-8. **^**Munch Pedersen, Jan (11 Sep 2006). "Known Perfect Numbers". Archived from the original on 2009-05-03. Retrieved 2009-09-16.**^**"Perfect Numbers". MIT. Retrieved 2009-09-16.**^**Chris Caldwell, "Mersenne Primes: History, Theorems and Lists" at The Prime Pages. Retrieved 2018-01-03.**^**The Penguin's Dictionary of curious and interesting numbers**^**Dickson, Leonard Eugene (1999-05-01).*Divisibility and primality*. p. 6. ISBN 9780821819340. Retrieved 2011-04-13.**^**Ochem, Pascal; Rao, Michael, "Odd Perfect Numbers Are Greater Than 10^1500",*MATHEMATICS OF COMPUTATION*, Volume 81, Number 279, July 2012, Pages 1869–1877. S 0025-5718(2012)02563-4. Article electronically published on January 30, 2012**^**"GIMPS Milestones Report". Retrieved 2018-08-04.