Wagstaff prime

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.

Examples

The first three Wagstaff primes are 3, 11, and 43 because

Known Wagstaff primes

The first few Wagstaff primes are:
Exponents which produce Wagstaff primes or probable primes are:

Generalizations

It is natural to consider more generally numbers of the form
where the base. Since for odd we have
these numbers are called "Wagstaff numbers base ", and sometimes considered a case of the repunit numbers with negative base.
For some specific values of, all are composite because of an "algebraic" factorization. Specifically, if has the form of a perfect power with odd exponent, then the fact that, with odd, is divisible by shows that is divisible by in these special cases. Another case is, with k a positive integer, where we have the aurifeuillean factorization.
However, when does not admit an algebraic factorization, it is conjectured that an infinite number of values make prime, notice all are odd primes.
For, the primes themselves have the following appearance: 9091, 909091, 909090909090909091, 909090909090909090909090909091, …, and these ns are: 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207,....
See Repunit#Repunit primes for the list of the generalized Wagstaff primes base.
The least primes p such that is prime are
The least bases b such that is prime are