mersenneforum.org Status of Wagstaff testing? and testing Mersenne primes for Wagstaff-ness
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2018-08-01, 20:10   #56
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,531 Posts

Quote:
 Originally Posted by diep Yet in our lifetime we do not yet know how far we will be able to test to find sufficient datapoints there.
Not going into this deep numerology staff, but we have already a lot of results, if I understand correctly the search for Wagstaff prime is complete up to p=15e6, the Mersenne prime search also reached this. We have 43 Wagtsaff prime, and **only** 39 Mersenne prime up to p=15e6, so what would you search? My 2 cents: probably the density is the same (a similar heuristic what used for Mersenne could work here), ofcourse the big drawback for Wagstaff that we have no (known) fast primality test.

2018-08-01, 20:15   #57
diep

Sep 2006
The Netherlands

30E16 Posts

Quote:
 Originally Posted by R. Gerbicz Not going into this deep numerology staff, but we have already a lot of results, if I understand correctly the search for Wagstaff prime is complete up to p=15e6, the Mersenne prime search also reached this. We have 43 Wagtsaff prime, and **only** 39 Mersenne prime up to p=15e6, so what would you search? My 2 cents: probably the density is the same (a similar heuristic what used for Mersenne could work here), ofcourse the big drawback for Wagstaff that we have no (known) fast primality test.
The first few n's, see it like Wagstaff is a propellor airplane. It had some advantages, was easier to build, yet Mersenne is a fighter jet. Lots of problems to get started and some hiccups, yet after it finally took off at the million bits area, it produces very systematic primes now.

Distance between each wagstaff and the next one simply is factor 3 nearly.

2018-08-01, 20:23   #58
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,531 Posts

Quote:
 Originally Posted by diep Distance between each wagstaff and the next one simply is factor 3 nearly.
Yeah, like at 13347311, 13372531.

 2018-08-01, 20:26 #59 diep     Sep 2006 The Netherlands 2·17·23 Posts You ignore what i wrote. To find the NEXT one, even if 3 prp's lurk nearby, you want to know the DISTANCE you need to search to find 'em. 3 x 13M = 40M
2018-08-02, 01:04   #60
GP2

Sep 2003

5×11×47 Posts

Quote:
 Originally Posted by diep most recent history: 13 / 4 = factor 3.25 distance 4 / 1 = factor 4.0 distance 981k / 374321 = factor 2.6 distance
If Wagstaff primes are distributed like Mersenne primes (is it true?), then we would expect an average ratio of 1.47576 between exponents of successive primes.

For Mersennes, the highest ratios are:

Code:
521 / 127 = 4.10
756839 / 216091 = 3.50
with everything else at 2.31 or lower.

For Wagstaffs, the highest ratios are:

Code:
1709 / 701 = 2.44
42737 / 14479 = 2.95
986191 / 374321 = 2.63
4031399 / 986191 = 4.09
13347311 / 4031399 = 3.31
with everything else at 2.02 or lower.

Those three very high ratios in a row do seem anomalous. But can we be certain they are real?

Maybe I'm mistaken, but I think the large rangers were tested with programs that assumed the Vrba-Reix conjecture is correct.

If there are counterexamples to this conjecture, then maybe some Wagstaff primes were missed?

Quote:
 Originally Posted by R. Gerbicz 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.
The previous testing was mostly done in 2013 and earlier.

Today mprime can do PRP testing with Gerbicz error checking, and it does not depend on Vrba-Reix being true. And we have faster CPUs with more cores, and faster GPUs that can find more factors.

I am seriously considering throwing some resources into Wagstaff testing old ranges using mprime. I don't know how far it would go, but the goal would be to publish the PRP residues and a consolidated list of new factors and old factors from multiple available sources, as a permanent verifiable record.

Sadly, there don't seem to be too many published results from the 2013 efforts, other than the primes that were found. Some lists of factors were published, but others are harder to track down and might even be lost.

Ideally, the PRP residues should be recorded with a lot more than 64 bits. If the only goal is double-check verifiability, then 64 bits is plenty; however, larger residues might let us do quick PRP-cofactor checking for newly discovered factors. (I presume the analysis at the linked page for Mersennes applies similarly for Wagstaffs, or is "inv" different?)

So, at least 512 bits, or with an eye to future decades, who knows, maybe even 2048. Storage is very cheap.

I'm not aware of any undoc.txt setting for mprime that would output residues larger than 64 bits, but hopefully it would be straightforward to add.

 2018-08-02, 10:10 #61 diep     Sep 2006 The Netherlands 2×17×23 Posts Yeah you are mistaken. Things at million bit size really works different than at a handful of bits. If you want to search in the dozens of millions bit range and then basing your expectation to find something upon a handful of bits for that, makes no sense at all. Last fiddled with by diep on 2018-08-02 at 10:19
2018-08-02, 11:22   #62
ET_
Banned

"Luigi"
Aug 2002
Team Italia

2×41×59 Posts

Quote:
 Originally Posted by diep Yeah you are mistaken. Things at million bit size really works different than at a handful of bits. If you want to search in the dozens of millions bit range and then basing your expectation to find something upon a handful of bits for that, makes no sense at all.
You consider an LL test passed or not by checking a single bit: true or false.

If a Law works in math, the number of involved bits does not matter.

 2018-08-02, 11:57 #63 diep     Sep 2006 The Netherlands 11000011102 Posts it's about the short term expectation how much of a computational work you need to throw in to find the next gem. Whether you find 1 or a million doesn't matter - you first have to do that huge computational effort to find at least 1. So the way how the PRP's have been spread also matter quite a lot. Last fiddled with by diep on 2018-08-02 at 12:01
 2018-08-02, 14:00 #64 paulunderwood     Sep 2002 Database er0rr 32·443 Posts If one was searching for Wagstaff co-factors then they can be PRP'ed over 2^p+1 before a final modular reduction. I guess Prime95 does this. Last fiddled with by paulunderwood on 2018-08-02 at 14:05
 2018-08-02, 14:33 #65 CRGreathouse     Aug 2006 3·1,993 Posts So in summary: diep: We should ignore everything but the big numbers, numbers at millions of bits aren't like numbers with just a few bits. The big Wagstaff exponents are spread very thin. GP2: You're throwing out most of the data, if you look at the broader picture you see the Wagstaff exponents aren't that thin. Also, the higher ranges might have missed some primes because searches relied on a conjecture.
 2018-08-02, 15:05 #66 diep     Sep 2006 The Netherlands 2×17×23 Posts You mean missed above 10M until 13M? Possible. Until 10M you won't find anything new. A 3-PRP will find anything of course there for both mersenne and wagstaff. Doing a 27-PRP test which the conjecture uses is pretty obvious the same thing like doing a 3-PRP at least. So it won't miss anything. Whether it is a method to prove them prime - that i leave up to the theoretists with enough time :)

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