Wagstaff prime

A Wagstaff primeMathworldPlanetmath p is a prime numberMathworldPlanetmath of the form 22n+1+13. The first few are 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, etc., given in A000979 of Sloane’s OEIS. The exponent of 2 in the formula given above must be an odd numberMathworldPlanetmathPlanetmath for the result to be an integer in the first place, and that exponent must not be composite.

Currently, the largest known Wagstaff prime, corresponding to an exponent of 42737, is approximately 4.383322622×1012864. As there is no known special primality test for Wagstaff primes, François Morain had to use elliptic curve primality proving (ECPP) over several months to prove the primality of this Wagstaff prime, finishing in August 2007. (Morain names these primes after Samuel Wagstaff, Jr.) A000978 lists nine exponents which give probable primesMathworldPlanetmath.


  • 1 François Morain, “Distributed primality proving and the primality of (23539+1)3Lecture Notes in Comput. Sci. 473 (1991): 110 - 123
Title Wagstaff prime
Canonical name WagstaffPrime
Date of creation 2013-03-22 17:42:33
Last modified on 2013-03-22 17:42:33
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41