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 2015 there are 48 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, 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
- ↑ 1.0 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.
- ↑ Munch Pedersen, Jan (11 Sep 2006). "Known Perfect Numbers". Retrieved 2009-09-16.
- ↑ "Perfect Numbers". MIT. Retrieved 2009-09-16.
- ↑ Chris Caldwell, "Mersenne Primes: History, Theorems and Lists" at The Prime Pages. Retrieved 2009-09-19.
- ↑ 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.
- ↑ "" Odd Perfect Numbers Are Greater Than 10^1500. Retrieved 2012-06-02.
- ↑ "". Retrieved 2014-11-16.