In an effort that ran until 2013, a group including Tony Reix, Vincent Diepeveen, Paul Underwood and Jeff Gilchrist tested all Wagstaff numbers below 10 million. In 2010 they

found one Wagstaff prime, with exponent 4031399.

This used the Vrba-Reix algorithm as implemented in the LLR software package by Jean PennĂ©. This is a heuristic method that is not mathematically proven.

Starting last August, I began double-checking this range and today the testing reached the 10 million milestone. As might have been expected, no new primes have been found.

Note this is not double-checking in the strict sense of verifying matching residues, since it uses the standard 3-PRP test and mprime software, updating to new versions and new builds as they become available. For exponents up to about 4.6 million the testing used a slightly-modified 29.4 version that printed out 2048-bit residues instead of 64 bits. Internally, the program has always used full-sized residues, so the algorithm was unchanged, it was merely a matter of printing out the extra bits. After this, the testing used mprime versions 29.5 and 29.6, which print 2048-bit residues as a standard feature.

These are "type 5" residues, which means the value calculated is

3^(2^p) mod (2^p + 1) and the least-significant 2048 bits are recorded. Since mprime uses Gerbicz error checking, there is reason to believe the results are reliable. As discussed

in another thread, prior to version 29.6 build 2 there was a flaw in mprime's implementation of Gerbicz error checking that could allow an undetected error to be introduced in the last few iterations on machines with unreliable hardware. However I am using c5 servers on AWS, which have ECC memory and a very strong track record of error-free results.

I am evaluating whether to continue double-checking beyond 10 million, since the computations are obviously getting more expensive. Currently axn is testing the 10.0 to 10.1 million range.

The next two Wagstaff primes with exponents 13347311 and 13372531

were found by Ryan Propper in 2013. He searched at least parts of the 10M, 11M, 12M and 13M ranges using 3-PRP testing and PFGW software, but it is not known whether the coverage was exhaustive. Unfortunately Ryan does not recall exactly which exponents he tested. Nonetheless my hunch at this point is that there are no new Wagstaff primes below 14 million.

A mini-website contains links to flat files with known factors and 2048-bit residues.