Inmathematics, theMersenne conjectures concern the characterization of a kind ofprime numbers calledMersenne primes, meaning prime numbers that are apower of two minus one.

The original, calledMersenne's conjecture, was a statement byMarin Mersenne in hisCogitata Physico-Mathematica (1644; see e.g. Dickson 1919) that the numbers${\displaystyle 2^n-1}$ were prime forn = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 (sequenceA109461 in theOEIS), and werecomposite for all other positiveintegersn ≤ 257. The first seven entries of his list (${\displaystyle 2^n-1}$ forn = 2, 3, 5, 7, 13, 17, 19) had already been proven to be primes by trial division before Mersenne's time;^[1] only the last four entries were new claims by Mersenne. Due to the size of those last numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as theLucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two entries are composite (those corresponding to the primesn = 67, 257) and three primes are missing (those corresponding to the primesn = 61, 89, 107). The correct list forn ≤ 257 is:n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

While Mersenne's originalconjecture is false, it may have led to theNew Mersenne conjecture.

TheNew Mersenne conjecture orBateman, Selfridge and Wagstaff conjecture (Bateman et al. 1989) states that for anyoddnatural numberp, if any two of the following conditions hold, then so does the third:

1. p = 2^k ± 1 orp = 4^k ± 3 for some natural numberk. ((sequenceA122834 in theOEIS))
2. 2^p − 1 is prime (aMersenne prime). ((sequenceA000043 in theOEIS))
3. (2^p + 1)/3 is prime (aWagstaff prime). ((sequenceA000978 in theOEIS))

Ifp is an odd composite number, then 2^p − 1 and (2^p + 1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the conjecture.

Currently, there are nine known numbers for which all three conditions hold: 3, 5, 7, 13, 17, 19, 31, 61, 127 (sequenceA107360 in theOEIS). Bateman et al. expected that no number greater than 127 satisfies all three conditions, and showed that heuristically no greater number would even satisfy two conditions, which would make the New Mersenne conjecture trivially true.

If at least one of thedouble Mersenne numbers MM61 and MM127 is prime, then the New Mersenne conjecture would be false, since both M61 and M127 satisfy the first condition (since they are Mersenne primes themselves), but (2^M61+1)/3 and (2^M127+1)/3 are both composite, they are divisible by 1328165573307087715777 and 886407410000361345663448535540258622490179142922169401, respectively.

As of 2025^[update], all the Mersenne primes up to 2^57885161 − 1 are known, and for none of these does the first condition or the third condition hold except for the ones just mentioned.^[2][3][4][5]Primes which satisfy at least one condition are

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 67, 79, 89, 101, 107, 127, 167, 191, 199, 257, 313, 347, 521, 607, 701, 1021, 1279, 1709, 2203, 2281, 2617, 3217, 3539, 4093, 4099, 4253, 4423, 5807, 8191, 9689, 9941, ... (sequenceA120334 in theOEIS)

Note that the two primes for which the original Mersenne conjecture is false (67 and 257) satisfy the first condition of the new conjecture (67 = 2⁶ + 3, 257 = 2⁸ + 1), but not the other two. 89 and 107, which were missed by Mersenne, satisfy the second condition but not the other two. Mersenne may have thought that 2^p − 1 is prime only ifp = 2^k ± 1 orp = 4^k ± 3 for some natural numberk, but if he thought it was "if and only if" he would have included 61.

The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false. However, according toRobert D. Silverman,John Selfridge agreed that the New Mersenne conjecture is "obviously true" as it was chosen to fit the known data andcounter-examples beyond those cases are exceedingly unlikely. It may be regarded more as a curious observation than as an open question in need ofproving.

Prime Pages shows that the New Mersenne conjecture is true for all integers less than or equal to 10000000^[2] by systematically listing all primes for which it is already known that one of the conditions holds. In fact, currently it is known that the New Mersenne conjecture is true for all integers less than or equal to the current search limit of the Mersenne primes (seethis page for the current search limit of the Mersenne primes), also currently it is known that the New Mersenne conjecture is true for all integers less than 1073741827 which satisfy the first condition, also currently it is known that the New Mersenne conjecture is true for all known integers which satisfy the second or third condition.^[3][4]

Lenstra,Pomerance, andWagstaff have conjectured that there are infinitely manyMersenne primes, and, more precisely, that the number of Mersenne primes less thanx isasymptotically approximated by

${\displaystyle e^{\gamma }\cdot {\log }_2{\log }_2(x),}$^[7]

where γ is theEuler–Mascheroni constant. In other words, the number of Mersenne primes with exponentp less thany is asymptotically

${\displaystyle e^{\gamma }\cdot {\log }_2(y).}$^[7]

This means that there should on average be about${\displaystyle e^{\gamma }\cdot {\log }_2(10)}$ ≈ 5.92 primesp of a given number of decimal digits such that${\displaystyle M_p}$ is prime. The conjecture is fairly accurate for the first 40 Mersenne primes (up to 2^20,996,011), but between 2^20,000,000 and 2^85,000,000 there are at least 12,^[8] rather than the expected number which is around 3.7.

More generally, the number of primesp ≤y such that${\displaystyle \frac{a^p-b^p}{a-b}}$ is prime (wherea,b arecoprime integers,a > 1, −a <b <a,a andb are not both perfectr-th powers for any natural numberr > 1, and −4ab is not a perfect fourth power) is asymptotically

${\displaystyle (e^{\gamma }+m\cdot {\log }_e(2))\cdot {\log }_a(y).}$

wherem is the largest nonnegative integer such thata and −b are both perfect 2^m-th powers. The case of Mersenne primes is one case of (a, b) = (2, 1).

• Gillies' conjecture on the distribution of numbers of prime factors of Mersenne numbers
• Lucas–Lehmer primality test
• Lucas primality test
• Catalan's Mersenne conjecture
• Mersenne's laws

• Bateman, P. T.;Selfridge, J. L.;Wagstaff Jr., Samuel S. (1989). "The new Mersenne conjecture".American Mathematical Monthly.96 (2). Mathematical Association of America:125–128.doi:10.2307/2323195.JSTOR 2323195.MR 0992073.
• Dickson, L. E. (1919).History of the Theory of Numbers. Carnegie Institute of Washington. p. 31.OL 6616242M. Reprinted by Chelsea Publishing, New York, 1971,ISBN 0-8284-0086-5.

• The Prime Glossary.New Mersenne prime conjecture.

• Conjectures about prime numbers
• Unsolved problems in number theory
• Mersenne primes

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