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 2016 there are 49 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.

The displayed ranks are among those perfect numbers which are known as of January 2016. 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. 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 2016-01-19.
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. 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
8. "GIMPS Milestones Report". Retrieved 2014-11-16.

External links

• Mersenne Primes: History, Theorems and Lists