Wagstaff prime
| Named after | Samuel S. Wagstaff, Jr. |
|---|---|
| Publication year | 1989 |
| Author of publication | Bateman, P. T., Selfridge, J. L., Wagstaff Jr., S. S. |
| No. of known terms | 43 |
| First terms | 3, 11, 43, 683 |
| Largest known term | (213372531+1)/3 |
| OEIS index | A000979 |
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 are related to 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:
As of October 2014[update], 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.
These numbers are proven to be prime for the values of q up to 83339. Those with q > 83339 are probable primes as of April 2015[ref]. 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 (Lucas-Lehmer-Riesel) 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.
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