20180721, 20:56  #1 
Sep 2003
101000011001_{2} Posts 
Status of Wagstaff testing? and testing Mersenne primes for Wagstaffness
Is there any centralized data repository for Wagstaff number testing? A list of exponents with factors, and a list of PRP test residues, possibly doublechecked?
Various people have tested extensively, but have the results been centralized anywhere? I think systematic testing has been done up to about 4M, and then Ryan Popper found (2^13372531+1)/3 and (2^13347311+1)/3 which are actually the largest known PRPs, and it's not clear if anyone has searched the gap between 4M and 13.3M However, I'd especially like to confirm that testing has been done to try to invalidate the New Mersenne Conjecture, by factoring or testing all the known Mersenne prime exponents larger than 4M as Wagstaff exponents, to see if any of them might be Wagstaff primes as well. I assumed this was the case, but maybe I'm wrong? Edit: Chris Caldwell's page says the Wagstaff status of Mersenne prime exponents 20996011 and 30402457 is unknown, with no info about any higher exponents. Is this still true? Last fiddled with by GP2 on 20180721 at 21:27 
20180721, 21:08  #2 
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Why not use my own error checking on these numbers? Even a single run is much stronger than your double or even triple/quadruple check if you compare only RES64. The same is true for Mersenne numbers.

20180721, 22:26  #3  
Sep 2003
5·11·47 Posts 
Quote:
Which programs can do Wagstaff PRP testing, and has your Gerbicz error check been implemented for them yet? Also, it's necessary to know which exponents p have known factors for (2^p + 1)/3 Last fiddled with by GP2 on 20180721 at 22:28 

20180721, 22:50  #4  
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Quote:
ps. If you have an algorithm/code for k*b^n+c, then you have a code also for (k*b^n+c)/d, because you need only one biginteger division (or even eliminate that also if d is "small"). 

20180721, 23:43  #5 
Sep 2003
2585_{10} Posts 

20180722, 07:56  #6 
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
OK, I was too compact/short.
Here are the deatils: we restrict to b=2 to enable my fast error checking let N=k*2^n+c (but we want a test for N/d), if N/d is a Fermat prp to base=a^d, then (a^d)^(N/d)==a^d mod (N/d), so a^N==a^d mod (N/d), for this first get res=a^(2^n) mod N using my check, then compute (you can assume that k,c is small, but this works in general) res2=res^k*a^c mod N, until this point you have not used d. Reduce this: res'=res2 mod (N/d) to get a^N mod (N/d), using a single biginteger division (or two if you count also the N/d division). [There is another option here, if you begin with a^k mod N, and then use squarings, in timing there is no real difference, also note that c<0 is possible]. Assumed that you are a super heavy p95 user (I'm not using it on a daily routine) , PRPCF doesn't really using the error checking for (2^p1)/d ? Your own problem (2^p+1)/3 is totally similar. In general I'm not checking things where I'm sure, but if it is really not in p95, then it should be. Have you seen this? http://www.mersenneforum.org/showthread.php?t=12157 Just picked up a random (known!) prime: (use the d flag) in worktodo.txt put PRP=2675,2,1350023,1 Code:
... [Work thread Jul 22 09:22] Resuming Gerbicz errorchecking PRP test of 2675*2^1350023+1 using allcomplex FMA3 FFT length 96K, Pass1=384, Pass2=256, clm=4, 4 threads ... [Work thread Jul 22 09:23] 2675*2^1350023+1 is a probable prime! Wg8: A97983C5,00000000 So at least for d=1 (or there f=1), it is really working. But for f>1 it is just throwing Code:
[Main thread Jul 22 09:28] Illegal line in worktodo.txt file: PRP=1,2,986191,1,3 Code:
[Work thread Jul 22 09:29] PRP test of 2^986191+1 aborted  number is divisible by 3 But if N=k*2^n+c is already working, then N/d should be also, the underlying Maths is so trivial. Last fiddled with by R. Gerbicz on 20180722 at 08:05 Reason: more info 
20180722, 08:10  #7 
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Bingo: you have to use this: PRP=1,2,986191,1,"3"
at least on Linux, and, then Code:
.. [Work thread Jul 22 10:07] Starting Gerbicz errorchecking PRP test of 2^986191+1/3 using allcomplex FMA3 FFT length 48K, Pass1=768, Pass2=64, clm=2, 4 threads [Work thread Jul 22 10:08] Iteration: 10000 / 986191 [1.01%], ms/iter: 0.124, ETA: 00:02:00 [Work thread Jul 22 10:08] Iteration: 20000 / 986191 [2.02%], ms/iter: 0.122, ETA: 00:01:57 .. [Work thread Jul 22 10:09] Iteration: 980000 / 986191 [99.37%], ms/iter: 0.122, ETA: 00:00:00 [Work thread Jul 22 10:10] 2^986191+1/3 is a probable prime! Wg8: D9C45894,00000000 Minor issue, but parentheses are missing here in text: 2^986191+1/3. Last fiddled with by R. Gerbicz on 20180722 at 08:18 
20180722, 10:14  #8 
Einyen
Dec 2003
Denmark
3,253 Posts 
I worked on the NMC:
http://mersenneforum.org/showthread.php?t=19229 http://hoegge.dk/mersenne/NMC.html But as you can see in that thread, most agree that it is a "joke" conjecture, so I have not worked on it since adding the newest Mersenne primes. You can trial factor Wagstaff numbers on GPU using mfaktcwin64.Wagstaff.exe Last fiddled with by ATH on 20180722 at 10:21 
20180722, 18:31  #9 
Sep 2003
101000011001_{2} Posts 
I was made aware of these resources that are useful for the New Mersenne Conjecture:
Factors of Wagstaff numbers with Mersenneprime exponents: http://bearnol.isageek.com/Mersenn...neplustwo.html PRP tests of Wagstaff numbers with Mersenneprime exponents: https://groups.google.com/forum/#!to...wo/j4RYTPh540 
20180722, 18:34  #10 
Sep 2003
5×11×47 Posts 

20180724, 08:24  #11 
Sep 2006
The Netherlands
2×17×23 Posts 
the short answer to all questions i got by email and messages and everything regarding wagstaff is YES.
Systematic testing from our part has been done until 10M. I have all results until 10M. TF i have to lookup how far i did do it, as i did do it in waves. I think i had it until 5254 bits done at about 2024 cores up to 12M15M, i would need to look that one up. For the later gpgpu TF software that is total peanuts of course. And YES of course Wagstaff has the same habits like Mersenne. The main differences between Wagstaff and Mersenne, is that there seems no official proving method for Wagstaff has been recognized. I know some French and a Canadian researcher think different there, yet i'm not gonna pick sides there and just cite what i know as a layman there. Another difference seems the factoring. Trial Factoring, but i lack hard data of Mersenne to compare it with, seems more effective for Wagstaff than it is for Mersenne. Some researcher who wants to take this important unpaid job on himself might be able to officially determine that. I just posted here a gutfeeling. Though there might be a simple explanation for it that has to do with the spread of primes there. Because we divide by 3, maybe it is the case there is less Wagstaffs between [n and 2n]. As a result TF seemingly is far more effective so it would be possible to search Wagstaff deeper than Mersenne using the same horse power. Yet you'll find less PRP's than Mersenne primes of course using that same horse power. I do not have a formula how many less PRP's there are between [n;2n] of Wagstaff versus Mersenne. Blindfolded i would guess it ranges somewhere between (log 3) / (log 2) and factor 3. That means you could be busy with it a long time and not find any PRP and run the risk that when you are close to next one, the Ryan Propper team then suddenly posts them online including a "we were so lucky letter", which Jeff Gilchrist and Tony Reix, when they had to run fast and were out of paper at the bathroom, used over there. 
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