List of perfect numbers

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The following is a list of the known perfect numbers, including 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 2013 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 1279 541625262…984291328 770 1952 Robinson
16 2203 108925835…453782528 1327 1952 Robinson
17 2281 994970543…139915776 1373 1952 Robinson
18 3217 335708321…628525056 1937 1957 Riesel
19 4253 182017490…133377536 2561 1961 Hurwitz
20 4423 407672717…912534528 2663 1961 Hurwitz
21 9689 114347317…429577216 5834 1963 Gillies
22 9941 598885496…073496576 5985 1963 Gillies
23 11213 395961321…691086336 6751 1963 Gillies
24 19937 931144559…271942656 12003 1971 Tuckerman
25 21701 100656497…141605376 13066 1978 Noll & Nickel
26 23209 811537765…941666816 13973 1979 Noll
27 44497 365093519…031827456 26790 1979 Nelson & Slowinski
28 86243 144145836…360406528 51924 1982 Slowinski
29 110503 136204582…603862528 66530 1988 Colquitt & Welsh
30 132049 131451295…774550016 79502 1983 Slowinski
31 216091 278327459…840880128 130100 1985 Slowinski
32 756839 151616570…565731328 455663 1992 Slowinski & Gage
33 859433 838488226…416167936 517430 1994 Slowinski & Gage
34 1257787 849732889…118704128 757263 1996 Slowinski & Gage
35 1398269 331882354…723375616 841842 1996 Armengaud, Woltman, et al.
36 2976221 194276425…174462976 1791864 1997 Spence, Woltman, et al.
37 3021377 811686848…022457856 1819050 1998 Clarkson, Woltman, Kurowski, et al.
38 6972593 955176030…123572736 4197919 1999 Hajratwala, Woltman, Kurowski, et al.
39 13466917 427764159…863021056 8107892 2001 Cameron, Woltman, Kurowski, et al.
40 20996011 793508909…206896128 12640858 2003 Shafer, Woltman, Kurowski, et al.
41 24036583 448233026…572950528 14471465 2004 Findley, Woltman, Kurowski, et al.
42 25964951 746209841…791088128 15632458 2005 Nowak, Woltman, Kurowski, et al.
43 30402457 497437765…164704256 18304103 2005 Cooper, Boone, Woltman, Kurowski, et al.
44 32582657 775946855…577120256 19616714 2006 Cooper, Boone, Woltman, Kurowski, et al.
45 37156667 204534225…074480128 22370543 2008 Elvenich, Woltman, Kurowski, et al.
46 42643801 144285057…377253376 25674127 2009 Strindmo, Woltman, Kurowski, et al.
47 43112609 500767156…145378816 25956377 2008 Smith, Woltman, Kurowski, et al.
48 57885161 169296395…270130176 34850340 2013 Cooper, Woltman, Kurowski, et al.

The displayed ranks are among those perfect numbers which are known as of February 2013. 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 23 February 2014 the search for Mersenne primes (and thereby even perfect numbers) became exhaustive up to the 43rd above.[8]

References[edit]

  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. ^ "GIMPS Milestones Report". Retrieved 2014-02-24.

External links[edit]