OEIS - A057732 : 2^n + 3

[T.Rex]

Originally Posted by Batalov

Are you sure it was not a simple ABC ($a*$b^$c$d)/$e test in llr?

http://www.primenumbers.net/prptop/detailprp.php?rank=1
(2^13372531+1)/3

Please also credit Vincent Diepeveen, Tony Reix, Jeff Gilchrist and Paul Underwood for their work on checking Wagstaff prime exponents!

(2^13372531+1)/3 is 3-PRP! (Ryan Propper, OpenPFGW)
(2^13372531+1)/3 is Vrba-Reix PRP! (Jeff Gilchrist, LLR)
(2^13372531+1)/3 is a probable prime! ("ATH", Prime95)
(2^13372531+1)/3 is Base 27 - Strong Fermat PRP! (Serge Batalov, LLR)
(2^13372531+1)/3 is Vrba-Reix PRP! (Serge Batalov, LLR)
(2^13372531+1)/3 is 5-PRP! (Serge Batalov, OpenPFGW)
(2^13372531+1)/3 is 7-PRP! (Serge Batalov, OpenPFGW)
(2^13372531+1)/3 is 11-PRP! (Serge Batalov, OpenPFGW)
(2^13372531+1)/3 is 13-PRP! (Serge Batalov, OpenPFGW)
(2^13372531+1)/3 is 17-PRP! (Serge Batalov, OpenPFGW)
(2^13372531+1)/3 is Lucas PRP! ("ATH", OpenPFGW)
(2^13372531+1)/3 is Lucas PRP! (Paul Underwood, OpenPFGW)
(L+2)^(N+1)==5 (mod N, L^2+1) (Paul Underwood, GMP)

[R.D. Silverman]

Originally Posted by T.Rex

Read Edouard Lucas books and papers, read Hugh C. Williams book about Lucas' work, read Ribenboim, read all the books and papers about this subject, read my papers, and then go back with a proof, so that you can prove you can do something useful rather than always saying that other people are nuts.
Have a good day.

Take your condescending attitude and fuck off. I did not say that you were nuts. I did not denigrate what you
are doing in any way. I DID say that the discussion was off-topic because it contained no math.
I did say that as the person proposing the methods it is YOUR job to provide the supporting arguments.

I *HAVE* read all of the books you suggest. In point of fact, I was asked by Hugh Williams to
read his book BEFORE he published. He asked me for comments. I still have the manuscipt.

YOU are the one promoting these ideas. I have no interest in them. They provide nothing beyond current MR tests.

BTW, you will not find a proof. I will offer a hint why: Your computations are but a specialized case
of the Frobenius methods developed by Grantham. They are nothing more than PRP tests.
If you had bothered to take my hint about specifying the group the computations work in you might understand why.
But you are too busy criticizing me for not doing YOUR job.

[Batalov]

Originally Posted by T.Rex

Remember that my algorithm Reix-Vrba for Wagstaff numbers has been used for finding the biggest known PRPs : (2^13372531+1)/3 and (2^13347311+1)/3 .

So, you are still saying that this was a correct statement?

[T.Rex]

Originally Posted by Batalov

So, you are still saying that this was a correct statement?

Hummmm Thinking deeper about it, I'm not so sure. I remember that Vincent, Jeff, Paul and I were trying to find a new big Wagstaff PRP when the guy said he found 2 new ones, without a word about how he did. I'm not sure he said how he did after...
So, my statement is not solid. Only a guess.
About performance, I'm not the guy who can answer. Though I used the tools, I never put my hands inside the code. And FFT is still magic for me !

[T.Rex]

Originally Posted by R.D. Silverman

... BTW, you will not find a proof. I will offer a hint why: Your computations are but a specialized case of the Frobenius methods developed by Grantham. They are nothing more than PRP tests. ...

As I said, I'm not a Mathematician. I can understand some things, but not complex stuff.
I'll read the paper:
FROBENIUS PSEUDOPRIMES by JON GRANTHAM
Anyway, sorry for being rude. But, if you think that these algorithms are not worth to study, please provide a Math explanation. Did you ever read papers about DiGraph properties and try to use them ?
My opinion is that it's weird about how easy it is to find LLT-like algorithms for finding PRPs for so many different kinds of numbers. For me, it is a signal that there is something solid here. To be studied. And, even if it does not lead to a proof, it's worth to work it. And you never know before digging.

[R.D. Silverman]

Originally Posted by T.Rex

As I said, I'm not a Mathematician. I can understand some things, but not complex stuff.
I'll read the paper:
FROBENIUS PSEUDOPRIMES by JON GRANTHAM
Anyway, sorry for being rude. But, if you think that these algorithms are not worth to study, please provide a Math explanation.

I have never said that they are not worth studying.

Did you ever read papers about DiGraph properties and try to use them ?

I don't NEED to. The properties of the computations you are doing are well explained by
the cyclic group decomposition of the twisted group in which you are working. Stop babbling
about "DiGraphs". Stop inventing your own notation.

My opinion is that it's weird about how easy it is to find LLT-like algorithms for finding PRPs for so many different kinds of numbers.

You are not entitled to an opinion until you study the math. You simply don't know enough.

For me, it is a signal that there is something solid here. To be studied. And, even if it does not lead to a proof, it's worth to work it. And you never know before digging.

You can "dig" all you want. What you are doing is well understand from a group-theoretic point of
view as well as from the Frobenius endomorphism of GF(N^2) Nothing else is needed.

These methods are not deterministic. This was established by Grantham.

They are nothing more than PRP tests. And, owing to the work of Pomerance and Kim, they are not
even as useful as a MR test. For the latter we have good estimates of the probability that the result
of a test is in error. However, the Pomerance/Kim analysis will not work for your tests.

Back in early 90's I had a discussion with Richard Pinch about applying the Pomerance/Kim techniques to the
LL setting. We decided that it would not work. In the P&K work, they estimate some sums, all of whose
terms are positive. For LL (working on the "plus" side of GF(N^2)) some terms are positive and some are
negative. The errors blow up and one can't get good estimates on the probabilities.

[T.Rex]

Originally Posted by R.D. Silverman

Back in early 90's I had a discussion with Richard Pinch about applying the Pomerance/Kim techniques to the
LL setting. We decided that it would not work. In the P&K work, they estimate some sums, all of whose
terms are positive. For LL (working on the "plus" side of GF(N^2)) some terms are positive and some are
negative. The errors blow up and one can't get good estimates on the probabilities.

Hmmm
"We decided". Do you mean that you built a proof ?
Or did you decide that it should not work because summing positive and negative terms led to a complex task and could not be estimated correctly ?
So, was your position based on a proof or on the opinion that it should not work ?

Open "Prime numbers" book by Crandall and Pomerance, second edition, pages 173 and 181. These 2 giant Math guys say in their book published in 2005 that: 1) there are tests for the numbers N such that N-1 is easy to factor, like Fermat numbers, using Pépin's test, and that 2) there are tests for the numbers N such that N+1 is easy to factor, like Mersenne numbers, using Lucas-Lehmer Test. However, today, it is clear (at least 3 proofs have been provided, from 1960 by Kustaa Inkeri till now by me and another guy) that the Pépin's test is the same as a Lucas-Lehmer Test, with a different seed. The same LLT technic applies to both these numbers, which are very different. In that case, these giant Math guys were just parrots, repeating what thousands of other parrots have repeated during years, without checking.

So, till someone writes a proof showing that using a cycle of the DiGraph under x^2-2 modulo N cannot be used as a primality test, but only as a PRP test, I do not believe you.
Up to now, we have never found a LLT-cycle pseudoprime. And the "Searching for Wagstaff PRP" team has checked up to: 17,500,000 (about 1,121,000 Wagstaff numbers checked). For sure, this is not a proof that there will never be a pseudoprime ! But it looks more and more probable that the Vrba-Reix technic used for finding Wagstaff PRPs is in fact a property of Wagstaff primes only, and thus a true primality test. Wait for a proof, anyway.

[schickel]

Incoming....

(Check the primality section. Not sure how long it's going to take to verify.)