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diep 2018-07-24 10:14

[QUOTE=GP2;492257]I was not really interested in Wagstaff testing until now, so I'm not familiar with it. Can mprime/Prime95 be used to do Wagstaff PRP testing? I was under the impression that it can only do k*b^n+c where all of these are integers.

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[/QUOTE]

Hello, we used LLR, which incorporates also the PRP-27 test that is considered by Tony Reix to give a proof it is a prime as well - yet for sure shows whether it's PRP.

I should still have Tony's paper there on proving Wagstaff.

A quick search here is not promising on this machine nor my mailbox in the bunker of xs4all seems to have lost most emails from Jeff Gilchrist. Bizarre.
Might need to boot some old machines to dig up some files there.

diep 2018-07-24 10:32

In old backup directory i found some of the TF results until 12M that i have.

Let me see how large of a zip file i am able to post on this forum. The TF results have been spreaded over dozens of files of course.

For todays gpu's it is peanuts to outdo this.

The PRP testing, later on prp-27 testing i didn't find all results yet.

diep 2018-07-24 10:36

And i see most has been TF'ed to 61 or 62 bits.

Not the 52-54 i remembered from head.

diep 2018-07-24 11:11

2 Attachment(s)
Attached a few zip files. Note i created the file with 7-zip so i hope it can also zip in compatible manner with zip and unzip if you select 'zip' there.

It has some of the TF results until 12M, mainly what has been done until november 2010 at the 16 core box and some older results from 2008 and 2009.

If you want to 'go further' it might be wise to just generate all primes and subtract every factored result after you verified the result.

Even though verification always has been turned on here when TF-ing the result one should not trust results here if you want to do a systematic attempt. Please do not blame me if there is an error.

I simply see too many files always on my machines that others in illegal manner have modified or altered. Would be interesting to know whether that happened with Wagstaff as well. The testing itself had been carried out with REG+ECC ram turned on, so the testing itself is not questionable. It's the hackers who nonstop modify stuff on my machines, despite quite paranoia security i made myself. Some of the input files that the 16 core box swallowed i cannot read - that's why i typed this.

The fact that i have linux firewalls and airgapped security of some sort doesn't seem to help all that.

Most has been TF'ed to 61 bits. At lower ranges it has been TF'ed to 62 bits.

If you start testing, you can safely start at 10M testing for PRP using LLR and the prp-27 test. Though i'm very D sure you won't find any prp up until 15M or as how far propper has systematically tested it all.
There is no records nor reliable statement there. So one might need to start from 10M.

We did test it all systematically until 10M. A couple of hundreds of double tests have been performed yet not at the last ranges that were all done by Jeff. Even though he's the only one who tested without ECC ram, and at an i7 which turboboosted a bit, i don't think there is any need for a double test other than test a handful Jeff did do nearby the 10M mark. This was all reliable folks who tested it.

Under 10M the only interesting test to perform is to generate all primes until 10M and subtract those from the TF results and the test results. A handful were removed by Jeff because of some P-1 testing which from my viewpoint was not worth the system time to be performed as TF works dramatically better than P-1 for Wagstaff. Though i have done this as well and Paul as well in some cases - it is always possible a few exponents were missed in this proces as i didn't use a database to determine that.

Attempts to try to use a database were there yet quickly abandonned. Public databases kind of suck for this. A few scripts is a better idea :)

The only reason 1 or 2 exponents might have been missed, is the fact that many files were prepared by hand. Even though combining them has been done with some utils i wrote (in C) a lot in the end has been done by hand. A mistake always can occur then. Odds we missed something there is very low though, as always checks at irregular times at a later period of time have been done to check all this.

Yet if you go do a systematic attempt you might want to redo all that.

A more problematic question right now is whether i can still locate all the PRP test results.

Note that for the core0..15 results, this was an ongoing TF effort, so possibly many exponents have not been sieved there.
As on my backup directory it is trivial the input files have been modified and some are unreadable now - i'm not so sure what to say about it.

Above 10M in short the only possible approach is to regenerate all primes, and subtract TF'ed results and happily redo the TF from scratch at what's left.

My suspicion is that it all has been systematically TF'ed until start 12M. The original input files run till 30M.
The "first wave" was TF'ing it all 61 bits deep which succeeded until 12M.

Do not bet money you find a new PRP between 10M and 30M though. Wagstaff is very unreliable formula there, not like the clockwork Mersenne and 3 * 2^n -1 are there.
You could hit it rich or could get buried one day without accomplishing the mission to find a new one.

If we do very bad form of extrapolation: 4M / 0.9 = 4.x and 13 / 4 = 3.25
If we carefully do 5 * 13 = 65M

You might need to search up to 65M to have some sort of assurance odds are reasonable ok there is 1 or a bunch of Wagstaffs within your search domain.

Batalov 2018-07-24 14:42

[QUOTE=diep;492382]I should still have Tony's paper there on proving Wagstaff.
Not proving, as we recall. These are all PRPs.

diep 2018-07-24 14:50

Maybe read what i wrote Batalov. He submitted it as a paper how to prove Wagstaffs. I'm not wasting my time reading details like that (gigantic prime is a gigantic prime to me whether it's industry grade or a proven prime no diff to me). It's up to others to decide whether it's enough to accept it as a proof or reject it.

Batalov 2018-07-24 14:55

[QUOTE=diep;492397]I'm not wasting my time reading details like that.[/QUOTE]
That's the spirit!

At least other people need to know and thank you for being honest about it.

diep 2018-07-24 15:00

I wouldn't have the time for it Batalov - busy releasing a 3d printer. Hope to make a billion dollars with it. Even though that wasn't the initial idea when i started with it :)

p.s. i do remember some communication of when i was a student though, that was in the year 2000 or before that on some usenet group. Not sure with whom. Maybe it was Caldwell.
He told me next: "just find first that huge industry grade prime then, after you got it, we'll find a way to prove it" :)

Batalov 2018-07-24 18:36

Surely no one has time to read.
Then this is a completely useless link - no one is going to read it anyway:

And in the strange alternative universe where no one has time to read, alternative facts will keep getting repeated as if they were truth, such as "some French and a Canadian researcher think different there", that "there is some secret cabal that makes proofs official" [B]but[/B] "the unofficial proofs are just as good".
In that universe
2 equals 4, because:
subtract 3 from both sides and square both sides ==>
1 = 1, which is "obviously true". The proofs in that quoted thread are [URL=""]only slightly different[/URL] from this logical paradox.
In that universe it doesn't matter that the [URL=""]proof was withdrawn[/URL]. "Some French and a Canadian researcher" will continue to think different, no matter what the facts are.

diep 2018-07-24 21:01

No worries about that Batalov - if my 3d printer fails i can still take that job as Trumps twitter advisor :)

GP2 2018-07-25 06:01

[QUOTE=diep;492387]A handful were removed by Jeff because of some P-1 testing which from my viewpoint was not worth the system time to be performed as TF works dramatically better than P-1 for Wagstaff.[/QUOTE]

It doesn't make sense that P−1 testing should be ineffective for Wagstaff.

Wagstaff numbers ( 2[SUP]p[/SUP] + 1) / 3 for odd p ) appear to share with Mersenne numbers ( 2[SUP]p[/SUP] − 1 ) the characteristic that all their factors are of the form 2kp + 1 for some k.

I don't have a number theory proof of this for Wagstaff, I just tested it empirically, for instance on the [URL=""]zip file that ATH posted[/URL], which has 1.75 million factors.

The fact that factors are of the form 2kp + 1 is what makes P−1 testing particularly effective.

If a Mersenne number has a prime factor f, then (f − 1) is an even composite number, and if (f − 1) is "smooth" (has lots of small factors), then P−1 testing has a good chance of detecting that factor f of the Mersenne number. But (f − 1) = 2 * p * k for some k, so (f − 1) is automatically already a little bit smooth, and it all depends on how smooth k is.

And that's why P−1 testing is particularly effective for Mersenne numbers, we get a head start on the quest for smoothness.

But Wagstaff numbers should share the exact same advantage.

However, I was digging up old Wagstaff-related threads, and came up with [URL=""]this one from 2011[/URL]. It seems that in the source code, the 2 * p efficiency boost for P−1 testing is only applied to Mersenne numbers. In other words, only for k*b^n+c where k = 1, b = 2 and c = −1.

I looked at the latest source code, for mprime/Prime95 version 29.4 b7 and it seems that it's unchanged from 2011, when the issue was first raised.

So I wonder if P−1 testing for Wagstaffs is inefficient because of a longstanding bug (or not extending a desirable feature to a broader range).

Can some number theory experts enlighten us? What can we say about the forms of the factors of k*b^n+c when the constants k, b and c take on various fixed values? I think George would need that information for certain before applying changes.

And if it is indeed possible to give a Mersenne-like boost to Wagstaff P−1 testing, it would be interesting to quickly compile a version with the modification. I'd run it and try to find new Wagstaff factors.

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