Mersenne Primes List

A Mersenne prime is a prime number of the form 2^p − 1, where the exponent p is prime. There are 52 known Mersenne primes. Their exponents begin 2, 3, 5, 7, 13 and extend to 136,279,841.

The largest known member is 2^136,279,841 − 1. It has 41,024,320 decimal digits. The list is short because a prime exponent is only a necessary condition: many prime values of p still produce a composite number.

52Known Mersenne primes

136,279,841Largest known exponent

41,024,320Digits in the largest known member

51* and 52*Provisional list positions

Complete List of Known Mersenne Primes

The table is ordered by exponent, not by discovery date. For very large entries, writing the full decimal expansion is impractical, so the standard form 2^p − 1 is used.

Known Mersenne primes ordered by exponent; 51* and 52* have provisional ordinal positions.
#	Exponent p	Mersenne prime	Digits	Discovered
1	2	2² − 1	1	c. 500 BCE
2	3	2³ − 1	1	c. 500 BCE
3	5	2⁵ − 1	2	c. 275 BCE
4	7	2⁷ − 1	3	c. 275 BCE
5	13	2¹³ − 1	4	1456
6	17	2¹⁷ − 1	6	1588
7	19	2¹⁹ − 1	6	1588
8	31	2³¹ − 1	10	1772
9	61	2⁶¹ − 1	19	1883
10	89	2⁸⁹ − 1	27	1911
11	107	2¹⁰⁷ − 1	33	1914
12	127	2¹²⁷ − 1	39	1876
13	521	2⁵²¹ − 1	157	1952
14	607	2⁶⁰⁷ − 1	183	1952
15	1,279	2^1,279 − 1	386	1952
16	2,203	2^2,203 − 1	664	1952
17	2,281	2^2,281 − 1	687	1952
18	3,217	2^3,217 − 1	969	1957
19	4,253	2^4,253 − 1	1,281	1961
20	4,423	2^4,423 − 1	1,332	1961
21	9,689	2^9,689 − 1	2,917	1963
22	9,941	2^9,941 − 1	2,993	1963
23	11,213	2^11,213 − 1	3,376	1963
24	19,937	2^19,937 − 1	6,002	1971
25	21,701	2^21,701 − 1	6,533	1978
26	23,209	2^23,209 − 1	6,987	1979
27	44,497	2^44,497 − 1	13,395	1979
28	86,243	2^86,243 − 1	25,962	1982
29	110,503	2^110,503 − 1	33,265	1988
30	132,049	2^132,049 − 1	39,751	1983
31	216,091	2^216,091 − 1	65,050	1985
32	756,839	2^756,839 − 1	227,832	1992
33	859,433	2^859,433 − 1	258,716	1994
34	1,257,787	2^1,257,787 − 1	378,632	1996
35	1,398,269	2^1,398,269 − 1	420,921	1996
36	2,976,221	2^2,976,221 − 1	895,932	1997
37	3,021,377	2^3,021,377 − 1	909,526	1998
38	6,972,593	2^6,972,593 − 1	2,098,960	1999
39	13,466,917	2^13,466,917 − 1	4,053,946	2001
40	20,996,011	2^20,996,011 − 1	6,320,430	2003
41	24,036,583	2^24,036,583 − 1	7,235,733	2004
42	25,964,951	2^25,964,951 − 1	7,816,230	2005
43	30,402,457	2^30,402,457 − 1	9,152,052	2005
44	32,582,657	2^32,582,657 − 1	9,808,358	2006
45	37,156,667	2^37,156,667 − 1	11,185,272	2008
46	42,643,801	2^42,643,801 − 1	12,837,064	2009
47	43,112,609	2^43,112,609 − 1	12,978,189	2008
48	57,885,161	2^57,885,161 − 1	17,425,170	2013
49	74,207,281	2^74,207,281 − 1	22,338,618	2016
50	77,232,917	2^77,232,917 − 1	23,249,425	2017
51*	82,589,933	2^82,589,933 − 1	24,862,048	2018
52*	136,279,841	2^136,279,841 − 1	41,024,320	2024

About the asterisks: positions 51 and 52 are provisional. There are 52 known Mersenne primes, but not every candidate between the verified 50th entry and the largest known exponent has completed the level of checking needed to lock those two ordinal positions.

Mersenne Prime Exponents

The exponent sequence is often the most useful way to store the list. Each value of p below produces a prime number when substituted into 2^p − 1:

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933, 136279841

A missing prime exponent does not mean the exponent itself is composite. It means the corresponding Mersenne number is composite. For example, 11 is prime, but 2¹¹ − 1 is not.

The First Mersenne Primes in Decimal Form

The earliest members are small enough to write normally. After that, the decimal expansions grow too long to be useful on a reference page.

The first 12 Mersenne primes written as decimal integers.
Exponent p	2^p − 1	Digits
2	3	1
3	7	1
5	31	2
7	127	3
13	8,191	4
17	131,071	6
19	524,287	6
31	2,147,483,647	10
61	2,305,843,009,213,693,951	19
89	618,970,019,642,690,137,449,562,111	27
107	162,259,276,829,213,363,391,578,010,288,127	33
127	170,141,183,460,469,231,731,687,303,715,884,105,727	39

Why a Prime Exponent Is Not Enough

If 2^p − 1 is prime, then p must be prime. A composite exponent can be ruled out immediately. If n = ab with both factors greater than 1, then:

2^ab − 1 = (2^a)^b − 1

An expression of the form x^b − 1 is divisible by x − 1, so a composite exponent forces the Mersenne number to be composite.

The reverse does not hold. A prime exponent only creates a candidate. The standard small example is:

p = 11 is prime
2¹¹ − 1 = 2047 = 23 × 89
2047 is composite, so it is not a Mersenne prime.

This is why the exponent list skips many ordinary primes. Values such as 11, 23 and 29 are prime exponents, but their Mersenne numbers are not prime.

How Many Digits Does a Mersenne Prime Have?

The number of decimal digits can be found without expanding 2^p − 1. For a Mersenne prime M_p = 2^p − 1, use:

digits = floor(p × log₁₀2) + 1

For p = 31, the calculation gives 10 digits, matching 2³¹ − 1 = 2,147,483,647. For p = 136,279,841, it gives 41,024,320 digits. The formula makes size comparisons practical even when the number itself is far too large to display.

How Fast the Known Mersenne Primes Grow

The early entries fit on a line of text. Later entries contain millions of digits. The growth comes from the exponential term 2^p, while the known exponents also spread farther apart as the search moves into larger ranges.

Selected size milestones from the known Mersenne prime list.
List position	Exponent p	Digits
12	127	39
24	19,937	6,002
32	756,839	227,832
38	6,972,593	2,098,960
48	57,885,161	17,425,170
50	77,232,917	23,249,425
51*	82,589,933	24,862,048
52*	136,279,841	41,024,320

The jump from the previous record at p = 82,589,933 to p = 136,279,841 added more than 16 million decimal digits. That scale is one reason Mersenne notation is far more useful than printing the integer itself.

The Largest Known Mersenne Prime

Largest known Mersenne prime

2^136,279,841 − 1

41,024,320 digits

This prime was found by Luke Durant on October 12, 2024 through the Great Internet Mersenne Prime Search. The initial probable-prime result used GpuOwl on an NVIDIA A100, and later checks confirmed primality. It remains the largest known prime number as well as the largest known Mersenne prime.

The previous record was 2^82,589,933 − 1, with 24,862,048 digits. The newer prime is therefore not a small extension of the old record; the difference in decimal length alone exceeds 16 million digits.

Why Positions 51 and 52 Are Provisional

The asterisk beside 51 and 52 is easy to misread. It does not mean those numbers are unconfirmed primes. Their primality has been confirmed. The uncertainty concerns their ordinal positions in the exponent-ordered list.

To declare a Mersenne prime permanently as number 51 or 52, smaller candidate exponents must be eliminated with enough verification. GIMPS reports that all exponents below 141 million have been tested at least once, while all tests below 81 million have been verified. Because the verification frontier is lower than the largest known exponents, an undiscovered Mersenne prime could still exist in the not-yet-fully-verified interval.

The useful distinction: there are 52 known Mersenne primes, but the final numerical rank of the two largest known entries is not yet locked.

List Position and Discovery Order Are Different

The number in the first column describes position by increasing exponent. It is not always the order in which the primes were discovered.

For example, 2¹²⁷ − 1 is listed as number 12, yet Édouard Lucas proved it prime in 1876. The smaller-exponent entries at p = 89 and p = 107 were discovered later, in 1911 and 1914. A similar effect appears in the computer era: p = 110,503 was discovered in 1988, while the larger p = 132,049 and p = 216,091 had already been found in 1983 and 1985.

This happens because a search does not always finish exponents in strict numerical order. Hardware, algorithms, independent verification and previously untested ranges can change the order in which results become known.

How a Candidate Reaches the Known List

Modern Mersenne searches start with prime exponents. Factor searches can remove many candidates before a full primality test is attempted. If no factor is found, a probable-prime test can examine the full Mersenne number.

A probable-prime result is then checked with a deterministic Mersenne-specific test such as the Lucas–Lehmer test. For p > 2, Lucas–Lehmer gives an exact primality criterion for 2^p − 1. This special structure is one reason Mersenne numbers dominate records for very large known primes.

For ordinary-size integers, the same prime/composite distinction can be checked directly with the Prime Number Checker. A number belongs on the Mersenne list only when it is both prime and exactly one less than a power of two with a prime exponent.

Search Progress Beyond the Current Record

The search does not stop at p = 136,279,841. Distributed testing continues above the current record while older results are independently verified. GIMPS has completed at least one test for every exponent below 141 million and has verified all tests below 81 million.

Those two frontiers serve different purposes. First-time testing searches new territory. Verification closes older gaps and is what eventually fixes provisional list positions. A new Mersenne prime could therefore come from a larger exponent, or from a lower exponent whose earlier result has not yet reached the required verification state.

Are There Infinitely Many Mersenne Primes?

No proof is known that the Mersenne prime sequence is infinite. There are infinitely many ordinary prime numbers, but that result does not imply that infinitely many primes have the special form 2^p − 1.

The known list therefore has an unusual status: it is complete for the primes discovered and confirmed so far, but no formula tells us the next exponent. The next Mersenne prime may lie close to the present search range or much farther away.

Mersenne Numbers, Candidates and Mersenne Primes

The difference between a Mersenne number, a candidate and a Mersenne prime.
Term	Condition	Example
Mersenne number	Any number of the form 2ⁿ − 1	2¹¹ − 1 = 2047
Mersenne prime candidate	2^p − 1 with prime p	2047, where p = 11
Mersenne prime	The candidate itself is prime	2¹³ − 1 = 8191

Every Mersenne prime has a prime exponent, but not every prime exponent produces a Mersenne prime. That single distinction explains why the exponent sequence is sparse and why each new entry must pass a full primality proof rather than a simple exponent check.