20130908, 20:01  #1 
Jun 2012
Boulder, CO
2×7×13 Posts 
New Wagstaff PRP exponents
Hello,
I believe I have found the two largest known Wagstaff primes, after Tony Reix's discovery of (2^4031399+1)/3. Here they are: Code:
(2^13347311 + 1)/3 is 3PRP! (16355.1659s+0.0028s) Code:
(2^13372531 + 1)/3 is 3PRP! (34165.4750s+0.0029s) Can anyone with some spare cycles help verify these using PFGW or other primality testing software? Thanks!  Ryan 
20130908, 20:37  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2×3^{3}×13^{2} Posts 
Congratulations!
You can: * try the VrbaReix as implemented in LLR (you have to modify llr.ini) * also run pfgw with b5, b7 (and another dozen bases). * and run pfgw with t, tp and tc. I'll run a few of these for you, in parallel. 
20130908, 21:16  #3 
Sep 2002
Database er0rr
2^{2}·23·37 Posts 
Wowee!

20130908, 21:18  #4 
Sep 2002
Database er0rr
2^{2}·23·37 Posts 
What ranges did you search, Ryan?

20130908, 23:17  #5 
Sep 2006
The Netherlands
677 Posts 
Finding 2 close to each other, similar like how things work in Mersenne sometimes,
Maybe the odds for finding a Wagstaff in range [n;2n] which seemed a diverging sequence, so odds getting slowly a tad less each doubling of n, maybe maybe this is odds it's converging towards a near similar chance like one has to find a Mersenne. Note that TF and P1 rates of Wagstaff are considerable better than for Mersenne, so when i say 'converging towards' i still mean a considerable worse chance in range [n;2n], yet not as bad as the real small odds it seemed like considering the previous 2 were just under a million and something in the 4 million bits. Moving towards the 4 million is factor 4+, then suddenly 2 at 13 million is factor 3, yet there is 2, where there is 2 there could be more. So that's pretty good news, of course assuming both are PRP! 
20130910, 00:20  #6 
Einyen
Dec 2003
Denmark
2×13×113 Posts 
Prime95 concur on the first one:
2^13347311+1/3 is a probable prime! We4: B33A699A,00000000 
20130910, 00:59  #7 
"Phil"
Sep 2002
Tracktown, U.S.A.
1,117 Posts 
A Hearty congratulations to the project! Thanks especially to Tony Reix, Paul Underwood, and Vincent Diepeveen for testing so many candidates over the years. I am surprised that it took until now for Five or Bust to lose its lock on the largest known probable prime (since January 2009), but apparently the Wagstaff search hit a fairly substantial gap.
If these were proven primes, they would come in at 11th and 12th largest known, just below the 39th Mersenne and just above the last discovery of Seventeen or Bust. Do you have any statistics supporting this? I'm not saying it is wrong, just that I would expect the rates to be very similar. 
20130910, 01:14  #8  
Sep 2006
The Netherlands
677_{10} Posts 
Quote:
Interesting is to know what range was checked and what TF was done to find these 2 and whether a similar algorithm was used that was used to find some of the largest Mersennes past few years, prioritizing which exponent to use based upon factorisation of N+1 and N1 of the exponent. As for the TF and P1, i tend to remember posts that Mersenne TF'ed roughly 50% and that P1 removed roughly 7.5%. Please correct that if it's different. For Wagstaff even a quick TF already removes 60% and with gpu's add another 10% and P1 removes also 10%. The overlap of deeper TF and P1 is not so large. So in total you look at a 7075% that gets removed pretty easily with far less computational effort than has been done for Mersenne with respect to the P1. More accurate statistics will be there in some months hopefully with gpu factorisation stats. It's been some years i datamined through Mersenne statistics there. Can't remember how shallow those were compared to what we're doing. Note that Mersenne is a 1 formula and that Wagstaff is a +1 formula which should already explain a lot. Mersenne give a reasonable steady number of primes in each given range just like 3 * 2^n  1 also does. Wagstaff so far was pretty much a gamble whether it would be converging or diverging or constant in odds to the next PRP. I would guess blindfolded doing factorisation attempts other than P1 to remove some composites from Wagstaffs list to be tested might be more succesful than for Mersenne. In all cases and with respect to any statement the important emphasis is on the word "similar". Even if something factors 1% better i would not consider that similar. 

20130910, 01:29  #9 
Jun 2003
Ottawa, Canada
491_{16} Posts 
I just finished both exponents with LLR and they also show as PRP:
Code:
(2^13347311+1)/3 is VrbaReix PRP! Time : 56958.781 sec. (2^13372531+1)/3 is VrbaReix PRP! Time : 57032.491 sec. 
20130910, 01:46  #10  
Jun 2012
Boulder, CO
2×7×13 Posts 
Quote:


20130910, 02:25  #11 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2×3^{3}×13^{2} Posts 
Note that for the pfgw trial factoring you would want to use something like this command line:
pfgw f{13372607*2,1} q(2^13372607+1)/3 (2^13372607+1)/3 has factors: 50253508240009 (this will only look for factors of form 2*p*k+1) Compare the running time of the above to this: pfgw f q(2^13372607+1)/3 ____________________________________ P.S. The VrbaReix tests and some LLR and PFGW tests in a bunch of bases on the same two exponents are almost done here, too. 
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