List of perfect numbers

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The following is a list of the known perfect numbers, and the exponents p that can be used to generate them (using the expression 2p−1× (2p − 1)) whenever 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 2019 there are 51 known perfect numbers in total.[2][3][4] For even perfect numbers, 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
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.
51 82,589,933 110847779864...007191207936 49,724,095 2018 Laroche, Woltman, Kurowski, Blosser, et al.

The displayed ranks are among those perfect numbers which are known as of December 2018. Some ranks may change later if smaller perfect numbers are discovered. It is known there is no odd perfect number below 101500 ≈ 24983.[6] GIMPS reported that by 8 April 2018 the search for Mersenne primes (and thereby even perfect numbers) became exhaustive up to the 47th above.[7]

References

  1. ↑ 1.0 1.1 1.2 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". http://amicable.homepage.dk/perfect.htm. 
  3. ↑ "Perfect Numbers". MIT. http://web.mit.edu/adorai/www/perfectnumbers.html. 
  4. ↑ Chris Caldwell, "Mersenne Primes: History, Theorems and Lists" at The Prime Pages. Retrieved 2018-01-03.
  5. ↑ The Penguin's Dictionary of curious and interesting numbers
  6. ↑ 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
  7. ↑ "GIMPS Milestones Report". Retrieved 2018-08-04.




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