New Mersenne conjecture

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In mathematics, the New Mersenne conjecture (or Bateman, Selfridge and Wagstaff conjecture) is a statement concerning certain prime numbers; it states that for any odd natural number p, if any two of the following conditions hold, then so does the third:

1. p = 2^k ± 1 or p = 4^k ± 3 for some natural number k.
2. 2^p − 1 is prime (a Mersenne prime).
3. (2^p + 1) / 3 is prime (a Wagstaff prime).

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 odd primes to verify the truth of the conjecture.

The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false.

Renaud Lifchitz has shown that the NMC is true up to 12,441,900 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.

References

• P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., Samuel S., The new Mersenne conjecture, Amer. Math. Monthly, 96 (1989) 125-128

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