20190319, 16:06  #1 
Sep 2003
101000010011_{2} Posts 
There are no new Wagstaff primes with exponent below 10 million
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 VrbaReix 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 doublechecking 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 doublechecking in the strict sense of verifying matching residues, since it uses the standard 3PRP 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 slightlymodified 29.4 version that printed out 2048bit residues instead of 64 bits. Internally, the program has always used fullsized 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 2048bit 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 leastsignificant 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 errorfree results. I am evaluating whether to continue doublechecking 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 3PRP 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 miniwebsite contains links to flat files with known factors and 2048bit residues. 
20190319, 16:19  #2 
Sep 2006
The Netherlands
677 Posts 
Many thanks for the report. I did TF above 10M  what nowadays with GPU's is way faster and deeper to achieve. Yet we didn't test a single exponent above 10mbit.
Of course you won't find anything new until those 2 PRP's that suddenly were posted. out of the blue. 
20190726, 18:21  #3 
May 2018
3^{3}·7 Posts 
Have you checked up to 13347311 yet? I would like to know if 13347311, 13372531 are really the next Wagstaff prime exponents.

20190727, 00:14  #4  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·7·11·59 Posts 
Quote:
Sometimes there are gaps and holes in largesize computations, and when someone gets around to fill up the gaps  a new prime is found. This happened in one of the PrimeGrid projects before, iirc. This also happened in GaussianMersenne search while closing a gap that was left undone. 

20190727, 08:52  #5  
Sep 2003
2,579 Posts 
Quote:
One issue is that GPUs like the recent Radeon VII have proven to be startlingly more costeffective for Mersenne testing than CPUbased testing, perhaps by up to an order of magnitude. If the gpuOwL program is eventually adapted to do Wagstaff testing in addition to Mersenne testing, then that would be the way to go, rather than using mprime. Sadly, the Radeon VII has already been discontinued, but there will surely be other suitable GPUs. So I'd rather wait and get more bang for the buck. 

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