# 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 2p−1× (2p − 1) where 2p − 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 2014 there are 48 known perfect numbers in total.[2][3][4]

Rank p Perfect number Digits Year Discoverer
1 2 6 1 Known to the Greeks[5]
2 3 28 2 Known to the Greeks
3 5 496 3 Known to the Greeks
4 7 8128 4 Known to the Greeks
5 13 33550336 8 1456 First seen in a medieval manuscript, Codex Lat. Monac.[6]
6 17 8589869056 10 1588 Cataldi[1]
7 19 137438691328 12 1588 Cataldi[1]
8 31 2305843008139952128 19 1772 Euler
9 61 265845599…953842176 37 1883 Pervushin
10 89 191561942…548169216 54 1911 Powers
11 107 131640364…783728128 65 1914 Powers
12 127 144740111…199152128 77 1876 Lucas
13 521 235627234…555646976 314 1952 Robinson
14 607 141053783…537328128 366 1952 Robinson
15 1,279 541625262…984291328 770 1952 Robinson
16 2,203 108925835…453782528 1,327 1952 Robinson
17 2,281 994970543…139915776 1,373 1952 Robinson
18 3,217 335708321…628525056 1,937 1957 Riesel
19 4,253 182017490…133377536 2,561 1961 Hurwitz
20 4,423 407672717…912534528 2,663 1961 Hurwitz
21 9,689 114347317…429577216 5,834 1963 Gillies
22 9,941 598885496…073496576 5,985 1963 Gillies
23 11,213 395961321…691086336 6,751 1963 Gillies
24 19,937 931144559…271942656 12,003 1971 Tuckerman
25 21,701 100656497…141605376 13,066 1978 Noll & Nickel
26 23,209 811537765…941666816 13,973 1979 Noll
27 44,497 365093519…031827456 26,790 1979 Nelson & Slowinski
28 86,243 144145836…360406528 51,924 1982 Slowinski
29 110,503 136204582…603862528 66,530 1988 Colquitt & Welsh
30 132,049 131451295…774550016 79,502 1983 Slowinski
31 216,091 278327459…840880128 130,100 1985 Slowinski
32 756,839 151616570…565731328 455,663 1992 Slowinski & Gage
33 859,433 838488226…416167936 517,430 1994 Slowinski & Gage
34 1,257,787 849732889…118704128 757,263 1996 Slowinski & Gage
35 1,398,269 331882354…723375616 841,842 1996 Armengaud, Woltman, et al.
36 2,976,221 194276425…174462976 1,791,864 1997 Spence, Woltman, et al.
37 3,021,377 811686848…022457856 1,819,050 1998 Clarkson, Woltman, Kurowski, et al.
38 6,972,593 955176030…123572736 4,197,919 1999 Hajratwala, Woltman, Kurowski, et al.
39 13,466,917 427764159…863021056 8,107,892 2001 Cameron, Woltman, Kurowski, et al.
40 20,996,011 793508909…206896128 12,640,858 2003 Shafer, Woltman, Kurowski, et al.
41 24,036,583 448233026…572950528 14,471,465 2004 Findley, Woltman, Kurowski, et al.
42 25,964,951 746209841…791088128 15,632,458 2005 Nowak, Woltman, Kurowski, et al.
43 30,402,457 497437765…164704256 18,304,103 2005 Cooper, Boone, Woltman, Kurowski, et al.
44 32,582,657 775946855…577120256 19,616,714 2006 Cooper, Boone, Woltman, Kurowski, et al.
45 37,156,667 204534225…074480128 22,370,543 2008 Elvenich, Woltman, Kurowski, et al.
46 42,643,801 144285057…377253376 25,674,127 2009 Strindmo, Woltman, Kurowski, et al.
47 43,112,609 500767156…145378816 25,956,377 2008 Smith, Woltman, Kurowski, et al.
48 57,885,161 169296395…270130176 34,850,340 2013 Cooper, Woltman, Kurowski, et al.

The displayed ranks are among those perfect numbers which are known as of November 2014. Some ranks may change later if smaller perfect numbers are discovered. It is known there is no odd perfect number below 101500.[7] GIMPS reported that by 8 November 2014 the search for Mersenne primes (and thereby even perfect numbers) became exhaustive up to the 44th above.[8]

## References

1. ^ a b c Crilly, Tony (2007). 50 mathematical ideas you really need to know. Quercus Publishing. p. 43. ISBN 978-1-84724-008-8.
2. ^ Munch Pedersen, Jan (11 Sep 2006). "Known Perfect Numbers". Retrieved 2009-09-16.
3. ^ "Perfect Numbers". MIT. Retrieved 2009-09-16.
4. ^ Chris Caldwell, "Mersenne Primes: History, Theorems and Lists" at The Prime Pages. Retrieved 2009-09-19.
5. ^ The Penguin's Dictionary of curious and interesting numbers
6. ^ Dickson, Leonard Eugene (1999-05-01). Divisibility and primality. p. 6. ISBN 9780821819340. Retrieved 2011-04-13.
7. ^ "[1]" Odd Perfect Numbers Are Greater Than 10^1500. Retrieved 2012-06-02.
8. ^ "[2]". Retrieved 2014-11-16.