A107360 Numbers p (necessarily prime) such that 2^p - 1 is a Mersenne prime and (2^p+1)/3 is a Wagstaff prime.

3, 5, 7, 13, 17, 19, 31, 61, 127

Offset: 1

Author

Lekraj Beedassy, May 23 2005

Keywords

• nonn
• hard
• more

Comments

Intersection of A000043 and A000978.

'The New Mersenne Conjecture' (Bateman-Selfridge-Wagstaff) states that if two of the following statements about an odd positive integer p are true, then the third one is also true: (a) p = 2^k +- 1 or p = 4^k +- 3, (b) 2^p - 1 is prime, (c) (2^p + 1)/3 is prime. (Amer Math Monthly, 96 (1989) p. 125.) - R. K. Guy, May 20 2005

The next term, if it exists, is not any currently known Mersenne prime exponent or Wagstaff prime exponent: it must be larger than A000043(48) = 57885161 and cannot be 74207281, 77232917, or 82589933. See Caldwell and both Wanless links. The New Mersenne Conjecture would require this sequence to be a subsequence of A122834, in which case the next term could not be less than A122834(28) = 2305843009213693951. See Caldwell and Höglund links. - Gord Palameta, Jun 28 2019, Jun 29 2024

p either has the form 2^k -+ 1 or the form 4^k -+ 3, according to the New Mersenne Conjecture. - Lekraj Beedassy, Sep 20 2006

Primes p such that (4^p - 1)/3 is a semiprime. - Arkadiusz Wesolowski, Jun 01 2013

Numbers m != 4 such that (4^m - 1)/3 is a semiprime. - Thomas Ordowski, Sep 25 2015

The indices of Wagstaff primes relating to the new Mersenne conjecture A122834 in a list of Jacobsthal numbers A001045. - Steve Homewood, Dec 01 2020

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 83.

Links

• P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125-128, 1989.
• Chris K. Caldwell, Prime Pages, The New Mersenne Prime Conjecture
• Andreas Höglund, New Mersenne Conjecture
• Carlos Rivera, Conjecture 15 - The New Mersenne Conjecture, The Prime Puzzles & Problems Connection.
• James Wanless, Mersenneplustwo Factorizations (for p in known Mersenne-prime exponents, all Wagstaff numbers have factors except p = 19937, 30402457, 74207281)
• James Wanless, posts to Google group "Mersenneplustwo" show that Wagstaff p = 30402457, 42643801 are composite by PRP test and Wagstaff p = 74207281 is composite by PRP test
• Wikipedia, New Mersenne conjecture

Crossrefs

Cf. A000043, A000978, A122834.

Programs

Magma

[p: p in PrimesUpTo(500) | IsPrime(2^p-1) and IsPrime((2^p+1) div 3)]; // Vincenzo Librandi, Sep 25 2015

Mathematica

Select[Prime@Range[31], PrimeQ[(2^# + 1)/3] && PrimeQ[2^# - 1] &] (* Arkadiusz Wesolowski, Jun 01 2013 *)

PARI

forprime(p=2, 1e3, if (!((2^p+1) % 3) && isprime((2^p+1)/3) && isprime(2^p-1), print1(p, ", "))); \\ Altug Alkan, Sep 25 2015