A000978 Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• hard
• nice
• more

Comments

It is easy to see that the definition implies that k must be an odd prime. - N. J. A. Sloane, Oct 06 2006

The terms from a(32) on only give probable primes as of 2018. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 10 2018

Prime numbers of the form 1+Sum_{i=1..m} 2^(2i-1). - Artur Jasinski, Feb 09 2007

There is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT): Let p be a prime integer > 3, N(p) = 2^p+1 and W(p) = N(p)/3, S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod N(p)). Then W(p) is prime iff S(p-1) == S(0) (mod W(p)). - Tony Reix, Sep 03 2007

As a member of the DUR team (Diepeveen, Underwood, Reix), and thanks to the LLR tool built by Jean Penne, I've found a new and big Wagstaff PRP: (2^4031399+1)/3 is Vrba-Reix PRP! This Wagstaff number has 1,213,572 digits and today is the 3rd biggest PRP ever found. I've done a second verification on a Nehalem core with the PFGW tool. - Tony Reix, Feb 20 2010

13347311 and 13372531 were found to be terms of this sequence (maybe not the next ones) by Ryan Propper in September 2013. - Max Alekseyev, Oct 07 2013

The next term is larger than 10 million. - Gord Palameta, Mar 22 2019

Ryan Propper found another likely term, 15135397, though it only corresponds to a probable prime. - Charles R Greathouse IV, Jul 01 2021

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. S. Wagstaff, Jr., personal communication.

Links

• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• C. Caldwell's The Top Twenty, Wagstaff.
• C. Caldwell, New Mersenne Conjecture
• H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
• H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
• Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer (annotated and scanned copy)
• J. E. Foster, Problem 174, A solution in primes, Math. Mag., 27 (1954), 156-157.
• R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
• R. K. Guy, Letter to N. J. A. Sloane, 1987
• H. Lifchitz, Mersenne and Fermat primes field
• H. & R. Lifchitz, PRP Top Records.
• Henri & Renaud Lifchitz, PRP Records.
• Gord Palameta, There are no new Wagstaff primes with exponent below 10 million, mersenneforum.org
• Ryan Propper et al., New Wagstaff PRP exponents, mersenneforum.org
• PRP top list: PRP top [From _Tony Reix_, Feb 20 2010]
• Tony Reix, Yahoo Primeform Group Message 10184 dd. Feb 20, 2010, reconstruction in html.
• T. Reix, Some Maths about the Vrba-Reix PRP test [From _Tony Reix_, Feb 20 2010]
• Djurre G. Sikkema, Probable primality testing for Wagstaff prime, Bachelor's project mathematics, Univ. Groningen (Netherlands 2024). See p. 32.
• S. S. Wagstaff, Jr., The Cunningham Project
• Eric Weisstein's World of Mathematics, Repunit
• Eric Weisstein's World of Mathematics, Wagstaff Prime
• Eric Weisstein's World of Mathematics, Integer Sequence Primes
• Wikipedia, Wagstaff prime
• R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Crossrefs

Cf. A107036 (indices of prime Jacobsthal numbers).

Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.

Programs

Haskell

a000978 n = a000978_list !! (n-1)
a000978_list = filter ((== 1) . a010051 . a001045) a065091_list
-- Reinhard Zumkeller, Mar 24 2013

Mathematica

Select[Range[5000], PrimeQ[(2^# + 1)/3] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Prime[Range[2,500]],PrimeQ[(2^#+1)/3]&] (* Harvey P. Dale, Jun 13 2022 *)

PARI

forprime(p=2,5000,if(ispseudoprime(2^p\/3),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

Python

from gmpy2 import divexact
from sympy import prime, isprime
A000978 = [p for p in (prime(n) for n in range(2,10**2)) if isprime(divexact(2**p+1,3))] # Chai Wah Wu, Sep 04 2014

Formula

a(n) = A107036(n) for n>1. - Alexander Adamchuk, Feb 10 2007

Extensions

a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix, Sep 03 2007

a(31)-a(39) from Robert G. Wilson v, Apr 11 2005

a(40) from Vincent Diepeveen (diep(AT)xs4all.nl) added by Alexander Adamchuk, Jun 19 2008

a(41) from Tony Reix, Feb 20 2010