Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 − 1. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 2 − 1 = 2047 = 23 × 89. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.

There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 × (2 − 1), where 2 − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of p = 2, 2 − 1 = 3 is prime, and 2 × (2 − 1) = 2 × 3 = 6 is perfect.

It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is (e / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a lower bound of 10.

The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2023, there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.

The displayed ranks are among indices currently known as of 2022; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of January 2024. The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last six digits of each number are shown.

Table of all 51 currently-known Mersenne primes and corresponding perfect numbers

Rank	p	Mersenne prime	Mersenne prime digits	Perfect number	Perfect number digits	Discovery	Discoverer	Method	Ref.
1	2	3	1	6	1	Ancient times	Known to Ancient Greek mathematicians	Unrecorded
2	3	7	1	28	2
3	5	31	2	496	3
4	7	127	3	8128	4
5	13	8191	4	33550336	8	1200s/c. 1456	Multiple	Trial division
6	17	131071	6	8589869056	10	1588	Pietro Cataldi
7	19	524287	6	137438691328	12
8	31	2147483647	10	230584...952128	19	1772	Leonhard Euler	Trial division with modular restrictions
9	61	230584...693951	19	265845...842176	37	November 1883	Ivan Pervushin	Lucas sequences
10	89	618970...562111	27	191561...169216	54	June 1911	Ralph Ernest Powers
11	107	162259...288127	33	131640...728128	65	June 1, 1914
12	127	170141...105727	39	144740...152128	77	January 10, 1876	Édouard Lucas
13	521	686479...057151	157	235627...646976	314	January 30, 1952	Raphael M. Robinson	LLT on SWAC
14	607	531137...728127	183	141053...328128	366
15	1,279	104079...729087	386	541625...291328	770	June 25, 1952
16	2,203	147597...771007	664	108925...782528	1,327	October 7, 1952
17	2,281	446087...836351	687	994970...915776	1,373	October 9, 1952
18	3,217	259117...315071	969	335708...525056	1,937	September 8, 1957	Hans Riesel	LLT on BESK
19	4,253	190797...484991	1,281	182017...377536	2,561	November 3, 1961	Alexander Hurwitz	LLT on IBM 7090
20	4,423	285542...580607	1,332	407672...534528	2,663
21	9,689	478220...754111	2,917	114347...577216	5,834	May 11, 1963	Donald B. Gillies	LLT on ILLIAC II
22	9,941	346088...463551	2,993	598885...496576	5,985	May 16, 1963
23	11,213	281411...392191	3,376	395961...086336	6,751	June 2, 1963
24	19,937	431542...041471	6,002	931144...942656	12,003	March 4, 1971	Bryant Tuckerman	LLT on IBM 360/91
25	21,701	448679...882751	6,533	100656...605376	13,066	October 30, 1978	Landon Curt Noll & Laura Nickel	LLT on CDC Cyber 174
26	23,209	402874...264511	6,987	811537...666816	13,973	February 9, 1979	Landon Curt Noll
27	44,497	854509...228671	13,395	365093...827456	26,790	April 8, 1979	Harry L. Nelson & David Slowinski	LLT on Cray-1
28	86,243	536927...438207	25,962	144145...406528	51,924	September 25, 1982	David Slowinski
29	110,503	521928...515007	33,265	136204...862528	66,530	January 29, 1988	Walter Colquitt & Luke Welsh	LLT on NEC SX-2
30	132,049	512740...061311	39,751	131451...550016	79,502	September 19, 1983	David Slowinski et al. (Cray)	LLT on Cray X-MP
31	216,091	746093...528447	65,050	278327...880128	130,100	September 1, 1985		LLT on Cray X-MP/24
32	756,839	174135...677887	227,832	151616...731328	455,663	February 17, 1992		LLT on Harwell Lab's Cray-2
33	859,433	129498...142591	258,716	838488...167936	517,430	January 4, 1994		LLT on Cray C90
34	1,257,787	412245...366527	378,632	849732...704128	757,263	September 3, 1996		LLT on Cray T94
35	1,398,269	814717...315711	420,921	331882...375616	841,842	November 13, 1996	GIMPS / Joel Armengaud	LLT / Prime95 on 90 MHz Pentium PC
36	2,976,221	623340...201151	895,932	194276...462976	1,791,864	August 24, 1997	GIMPS / Gordon Spence	LLT / Prime95 on 100 MHz Pentium PC
37	3,021,377	127411...694271	909,526	811686...457856	1,819,050	January 27, 1998	GIMPS / Roland Clarkson	LLT / Prime95 on 200 MHz Pentium PC
38	6,972,593	437075...193791	2,098,960	955176...572736	4,197,919	June 1, 1999	GIMPS / Nayan Hajratwala	LLT / Prime95 on IBM Aptiva with 350 MHz Pentium II processor
39	13,466,917	924947...259071	4,053,946	427764...021056	8,107,892	November 14, 2001	GIMPS / Michael Cameron	LLT / Prime95 on PC with 800 MHz Athlon T-Bird processor
40	20,996,011	125976...682047	6,320,430	793508...896128	12,640,858	November 17, 2003	GIMPS / Michael Shafer	LLT / Prime95 on Dell Dimension PC with 2 GHz Pentium 4 processor
41	24,036,583	299410...969407	7,235,733	448233...950528	14,471,465	May 15, 2004	GIMPS / Josh Findley	LLT / Prime95 on PC with 2.4 GHz Pentium 4 processor
42	25,964,951	122164...077247	7,816,230	746209...088128	15,632,458	February 18, 2005	GIMPS / Martin Nowak
43	30,402,457	315416...943871	9,152,052	497437...704256	18,304,103	December 15, 2005	GIMPS / Curtis Cooper & Steven Boone	LLT / Prime95 on PC at University of Central Missouri
44	32,582,657	124575...967871	9,808,358	775946...120256	19,616,714	September 4, 2006
45	37,156,667	202254...220927	11,185,272	204534...480128	22,370,543	September 6, 2008	GIMPS / Hans-Michael Elvenich	LLT / Prime95 on PC
46	42,643,801	169873...314751	12,837,064	144285...253376	25,674,127	June 4, 2009	GIMPS / Odd Magnar Strindmo	LLT / Prime95 on PC with 3 GHz Intel Core 2 processor
47	43,112,609	316470...152511	12,978,189	500767...378816	25,956,377	August 23, 2008	GIMPS / Edson Smith	LLT / Prime95 on Dell OptiPlex PC with Intel Core 2 Duo E6600 processor
48	57,885,161	581887...285951	17,425,170	169296...130176	34,850,340	January 25, 2013	GIMPS / Curtis Cooper	LLT / Prime95 on PC at University of Central Missouri
*	68,684,599	Lowest unverified milestone
49	74,207,281	300376...436351	22,338,618	451129...315776	44,677,235	January 7, 2016	GIMPS / Curtis Cooper	LLT / Prime95 on PC with Intel Core i7-4790 processor
50	77,232,917	467333...179071	23,249,425	109200...301056	46,498,850	December 26, 2017	GIMPS / Jonathan Pace	LLT / Prime95 on PC with Intel Core i5-6600 processor
51	82,589,933	148894...902591	24,862,048	110847...207936	49,724,095	December 7, 2018	GIMPS / Patrick Laroche	LLT / Prime95 on PC with Intel Core i5-4590T processor
*	117,378,311	Lowest untested milestone

Historically, the largest known prime number has often been a Mersenne prime.