Mersenne conjectures

In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.

Original Mersenne conjecture

The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physica-Mathematica (1644; see e.g. Dickson 1919) that the numbers $2^{n}-1$ 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 = 2^k ± 1 or p = 4^k ± 3 for some natural number k. ( A122834)
2^p − 1 is prime (a Mersenne prime). ( A000043)
(2^p + 1) / 3 is prime (a Wagstaff prime). ( A000978)

If p 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, 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 2^p - 1 is prime if and only if p = 2^k ± 1 or p = 4^k ± 3 for some natural number k.

status for Mersenne conjecture for the first 100 primes
2	3	5	7	11	13	17	19	23	29
31	37	41	43	47	53	59	61	67	71
73	79	83	89	97	101	103	107	109	113
127	131	137	139	149	151	157	163	167	173
179	181	191	193	197	199	211	223	227	229
233	239	241	251	257	263	269	271	277	281
283	293	307	311	313	317	331	337	347	349
353	359	367	373	379	383	389	397	401	409
419	421	431	433	439	443	449	457	461	463
467	479	487	491	499	503	509	521	523	541

p is of the form 2ⁿ±1 or 4ⁿ±3	2^p-1 is prime	(2^p+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 20,996,010^[1] by systematically testing all odd 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–Wagstaff 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

e^{\gamma }\cdot \log _{2}\log(x),

^[2]

where γ is the Euler–Mascheroni constant. In other words, the number of Mersenne primes with exponent p less than y is asymptotically

e^{\gamma }\cdot \log _{2}(y).

^[2]

This means that there should on average be about $e^{\gamma }\cdot \log _{2}(10)$ ≈ 5.92 primes p of a given number of decimal digits such that $M_{p}$ is prime.

References

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.
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
1 2 Heuristics: Deriving the Wagstaff Mersenne Conjecture. The Prime Pages. Retrieved on 2014-05-11.

External links

The Prime Pages. Mersenne's conjecture.

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[1] The New Mersenne Prime Conjecture on Prime Pages

[heur-2] 1 2 Heuristics: Deriving the Wagstaff Mersenne Conjecture. The Prime Pages. Retrieved on 2014-05-11.