A107360 - OEIS

%I #101 Jul 02 2024 08:21:44

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

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

%C Intersection of A000043 and A000978.

%C '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

%C 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

%C 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

%C Primes p such that (4^p - 1)/3 is a semiprime. - _Arkadiusz Wesolowski_, Jun 01 2013

%C Numbers m != 4 such that (4^m - 1)/3 is a semiprime. - _Thomas Ordowski_, Sep 25 2015

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

%H P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., <a href="http://www.jstor.org/stable/2323195">The New Mersenne Conjecture</a>, Amer. Math. Monthly 96, 125-128, 1989.

%H Chris K. Caldwell, Prime Pages, <a href="https://primes.utm.edu/mersenne/NewMersenneConjecture.html">The New Mersenne Prime Conjecture</a>

%H Andreas Höglund, <a href="http://www.hoegge.dk/mersenne/NMC.html">New Mersenne Conjecture</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_015.htm">Conjecture 15 - The New Mersenne Conjecture</a>, The Prime Puzzles & Problems Connection.

%H James Wanless, <a href="https://sites.google.com/site/bearnol/math/mersenneplustwo">Mersenneplustwo Factorizations</a> (for p in known Mersenne-prime exponents, all Wagstaff numbers have factors except p = 19937, 30402457, 74207281)

%H James Wanless, posts to Google group "Mersenneplustwo" show that <a href="https://groups.google.com/forum/#!topic/Mersenneplustwo/j4RYTPh54-0">Wagstaff p = 30402457, 42643801 are composite by PRP test</a> and <a href="https://groups.google.com/forum/#!topic/Mersenneplustwo/PJ3e9Rk1DuQ">Wagstaff p = 74207281 is composite by PRP test</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/New_Mersenne_conjecture">New Mersenne conjecture</a>

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

%o (Magma) [p: p in PrimesUpTo(500) | IsPrime(2^p-1) and IsPrime((2^p+1) div 3)]; // _Vincenzo Librandi_, Sep 25 2015

%o (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

%Y Cf. A000043, A000978, A122834.

%K nonn,hard,more

%O 1,1

%A _Lekraj Beedassy_, May 23 2005