In number theory, a Wagstaff prime is a prime number of the form where p is an odd prime.
Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference.
Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography.
The first three Wagstaff primes are 3, 11, and 43 because The first few Wagstaff primes are: Exponents which produce Wagstaff primes or probable primes are: It is natural to consider[2] more generally numbers of the form where the base
odd we have these numbers are called "Wagstaff numbers base
", and sometimes considered[3] a case of the repunit numbers with negative base
) are composite because of an "algebraic" factorization.
has the form of a perfect power with odd exponent (like 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, 729, 1000, etc.
(sequence A070265 in the OEIS)), then the fact that
(sequence A141046 in the OEIS)), where we have the aurifeuillean factorization.
does not admit an algebraic factorization, it is conjectured that an infinite number of
prime, notice all
are odd primes.
, the primes themselves have the following appearance: 9091, 909091, 909090909090909091, 909090909090909090909090909091, … (sequence A097209 in the OEIS), and these ns are: 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... (sequence A001562 in the OEIS).
See Repunit#Repunit primes for the list of the generalized Wagstaff primes base
(Generalized Wagstaff primes base
are generalized repunit primes base
is prime are (starts with n = 2, 0 if no such p exists) The least bases b such that
is prime are (starts with n = 2)