Wagstaff prime


In number theory, a Wagstaff prime is a prime number p of the form
where q 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:
, known exponents which produce Wagstaff primes or probable primes are:
In February 2010, Tony Reix discovered the Wagstaff probable prime:
which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.
In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:
and
Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.
Note that when p is a Wagstaff prime, need not to be prime, first counterexample is p=683, and it is conjectured that if p is a Wagstaff prime and p>43, then is composite.

Primality testing

Primality has been proven or disproven for the values of q up to 83339. Those with q > 83339 are probable primes. The primality proof for q = 42737 was performed by François Morain in 2007 with a distributed ECPP implementation running on several networks of workstations for 743 GHz-days on an Opteron processor. It was the third largest primality proof by ECPP from its discovery until March 2009.
Currently, the fastest known algorithm for proving the primality of Wagstaff numbers is ECPP.
The LLR tool by Jean Penné is used to find Wagstaff probable primes by means of the Vrba-Reix test. It is a PRP test based on the properties of a cycle of the digraph under x^2-2 modulo a Wagstaff number.

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 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 for the list of the generalized Wagstaff primes base.
Least prime p such that is prime are
Least base b such that is prime are