Original Mersenne conjecture
The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physico-Mathematica (1644; see e.g. Dickson 1919) that the numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n ≤ 257. Due to the size of these 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 the Lucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two are composite (n = 67, 257) and three omitted primes (n = 61, 89, 107). The correct list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
While Mersenne's original conjecture is false, it may have led to the New Mersenne conjecture.
New Mersenne conjecture
The New Mersenne conjecture or Bateman, Selfridge and Wagstaff conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third:
- p = 2k ± 1 or p = 4k ± 3 for some natural number k. (OEIS: A122834)
- 2p − 1 is prime (a Mersenne prime). (OEIS: A000043)
- (2p + 1) / 3 is prime (a Wagstaff prime). (OEIS: A000978)
Currently, the known numbers for which all three conditions hold are: 3, 5, 7, 13, 17, 19, 31, 61, 127 (sequence A107360 in the OEIS). It is also a conjecture that no number which is greater than 127 holds all three conditions.
Primes which hold 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, ... (sequence A120334 in the OEIS)
Note that the two primes which Mersenne makes fault (67 and 257) are both in the conjecture (67=26+3, 257=28+1), but 89 and 107 are not. Thus, originally, Mersenne may have thought that 2p − 1 is prime if and only if p = 2k ± 1 or p = 4k ± 3 for some natural number k.[dubious ]
|p is of the form 2n±1 or 4n±3||2p-1 is prime||(2p+1)/3 is prime||p holds at least one condition|
The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false. However, according to Robert D. Silverman, John Selfridge agreed that the New Mersenne conjecture is "obviously true" as it was chosen to fit the known data and counter-examples beyond those cases are exceedingly unlikely. It may be regarded more as a curious observation than as an open question in need of proving.
Renaud Lifchitz has shown that the NMC is true for all integers less than or equal to 30,402,456 by systematically testing all primes for which it is already known that one of the conditions holds. His website documents the verification of results up to this number. Another, currently more up-to-date status page on the NMC is The New Mersenne Prime conjecture.
Lenstra, Pomerance, and Wagstaff have conjectured that there is an infinite number of Mersenne primes, and, more precisely, that the number of Mersenne primes less than x is asymptotically approximated by
where γ is the Euler–Mascheroni constant. In other words, the number of Mersenne primes with exponent p less than y is asymptotically
This means that there should on average be about ≈ 5.92 primes p of a given number of decimal digits such that is prime. The conjecture is fairly accurate for the first 40 Mersenne primes, but between 220,000,000 and 285,000,000 there are at least 12, rather than the expected number which is around 3.7.
More generally, the number of primes p ≤ y such that is prime (where a, b are coprime integers, a > 1, −a < b < a, a and b are not both perfect r-th powers for any natural number r > 1, and −4ab is not a perfect fourth power) is asymptotically
where m is the largest nonnegative integer such that a and −b are both perfect 2m-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. Mathematical Association of America. 96 (2): 125–128. doi:10.2307/2323195. JSTOR 2323195. MR 0992073.CS1 maint: multiple names: authors list (link)
- 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 New Mersenne Prime Conjecture on Prime Pages
- Heuristics: Deriving the Wagstaff Mersenne Conjecture. The Prime Pages. Retrieved on 2014-05-11.
- Michael Le Page (Aug 10, 2019). "Inside the race to find the first billion-digit prime number". New Scientist.
- The Prime Pages. Mersenne's conjecture.