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2010-02-24, 21:43   #23
T.Rex

Feb 2004
France

13·71 Posts
Nice !

Quote:
 Originally Posted by Cybertronic short addition :
Seems you have found a cycle and a seed. It gives 14 when q%2=2 and 194 when q%3=1.

You should use gp/PARI !
s=14;q=17;W=(2^q+1)/3;S=s;for(i=1,q-2,S=Mod(S^2-2,W);print(S))

Interesting...

Thanks !

Tony

 2010-02-24, 22:09 #24 Cybertronic     Jan 2007 Germany 421 Posts Thanks! You have right: p mod 3=2 -> S=14 , p mod 3=1 -> S=194 Curios, for both numbers (2^p+1)/3 and (2^p+1) work this program correct. It is not an accident, it is an coherence.
2010-02-24, 22:18   #25
T.Rex

Feb 2004
France

13·71 Posts

Quote:
 Originally Posted by Cybertronic Curios, for both numbers (2^p+1)/3 and (2^p+1) work this program correct. It is not an accident, it is an coherence.
Yes. That can be proved.

Continue to search. If you find a universal (for all Wagstaff primes) seed that leads to a cycle of la ength that divides q-1, I'm very interested !!

Tony

 2010-02-24, 22:40 #26 ixfd64 Bemusing Prompter     "Danny" Dec 2002 California 97C16 Posts Even though this is only a conjecture at the moment, it is still a major milestone in number theory. I'm surprised that no academic sources have picked up on it yet. I e-mailed MathWorld and Chris Caldwell but never got any response. Anyone else want to try?
 2010-02-25, 04:32 #27 Cybertronic     Jan 2007 Germany 1A516 Posts Like I said, me is it known since over 10 years but it was declined. I can't proved it, I have to little know how. It is NOT a trivial fermat test, still it is true for wagstaff primes. AND ! It is also for prime of form 2^n+3 : Here we have more exponents, because not only a prime. 10 P=2 20 N=2^P+3 30 T=1:S=14 40 S=(S^2 mod N)-2 50 inc T 60 if T=P then goto 100 70 goto 40 100 if or{S=(14 mod N),S=(194 mod N)} then print P; 120 P=P+1:goto 20 run 3 4 6 7 12 15 16 18 28 30 55 67 84 228 390 784 1110 1704 2008 2139 2191 2367 2370 Break in 40 Last fiddled with by Cybertronic on 2010-02-25 at 04:37
2010-02-25, 11:06   #28
T.Rex

Feb 2004
France

13×71 Posts
Ned help...

Quote:
 Originally Posted by ixfd64 Even though this is only a conjecture at the moment, it is still a major milestone in number theory. I'm surprised that no academic sources have picked up on it yet. I e-mailed MathWorld and Chris Caldwell but never got any response. Anyone else want to try?
Samuel Wagstaff was interested in Vrba conjecture and he asked a student to work on it, with no success.
HC Williams was also interested in my attempt to prove my conjecture for Mersenne Cycles (a first step before Vrba conjecture) and he gave me some help. I have put on my site a paper summarizing the ideas I've explored (still with the Ribenboim/Williams techniques, based on Lucas series, but with an idea of using the period of a special Lucas sequence. And it may be all nuts...).

I think that there is an interesting study to be done on the properties of the Cycles of a DiGraph for Mersennes, Wagstaff and Fermat numbers. Some kind of continuing the work done by Shallit&Vasiga. By experimentations, I've found interesting properties dealing with Cycles of the DiGraph under x^2-2 modulo a Mersenne prime that some other people have proven. More should be discovered.

What is the most important, I think, is:
1) to prove an efficient primality test for numbers that are not exactly N+1 or N-1 (with N easily factorizable), like Wagstaff numbers are, and
2) to explore the possibility to prove that a number is NOT prime must faster than with LLT or with other methods (except finding a factor !!) by using the property of a Cycle that divides q-2. My hope is that it will help to prove that a Fermat number is NOT prime much faster than searching by chance for a factor or by running Pépin's test (which would require years for last status-unknown candidate).

Moreover, this technique (use Cycles or Trees) can be extended to other kinds of numbers and with other functions than x^2-2. Edouard Lucas found some tests of this kind, but without (as usual for him !) providing a perfect modern proof that is today required. If Lucas had found this new Wagstaff PRP, he would have said: It's a prime ! like he did for other Mersenne primes, at a time he had not provided a full proof, before Lehmer came.

Tony

 2010-02-25, 13:08 #29 philmoore     "Phil" Sep 2002 Tracktown, U.S.A. 3·373 Posts Your number now is listed on the Lifchitz site: http://www.primenumbers.net/prptop/prptop.php The third mega-digit prp - congratulations, and go find another!
2010-02-25, 20:01   #30
T.Rex

Feb 2004
France

13×71 Posts

Quote:
 Originally Posted by philmoore Your number now is listed on the Lifchitz site: http://www.primenumbers.net/prptop/prptop.php The third mega-digit prp - congratulations, and go find another!
Good !
Yes ! With mountains, there is a limit... But, with PRPs or primes, there is no limit ! So, yes, already looking for another one ! One more year or more to wait for a new one ?! :)
Tony

 2010-02-25, 20:31 #31 axn     Jun 2003 3×5×347 Posts At the risk of derailing the thread, the V-R test can be generalized for non-base-2 primes (both for generalized repunits (b^n-1)/(b-1) and generalized wagstaff (b^n+1)/(b+1))
2010-02-25, 20:36   #32
T.Rex

Feb 2004
France

92310 Posts

Quote:
 Originally Posted by axn At the risk of derailing the thread, the V-R test can be generalized for non-base-2 primes (both for generalized repunits (b^n-1)/(b-1) and generalized wagstaff (b^n+1)/(b+1))
Good !!! Do you have examples for generalized Wagstaff ?
Tony

2010-02-26, 01:25   #33
axn

Jun 2003

3·5·347 Posts

Quote:
 Originally Posted by T.Rex Good !!! Do you have examples for generalized Wagstaff ? Tony
For base 3:
Define f(x) = x^3-3*x
Test for N=(3^p-1)/(3-1): s0=4, si+1=f(si), sp$\eq$s1 (mod N)
Test for N=(3^p+1)/(3+1): s0=8, si+1=f(si), sp$\eq$s1 (mod N)

For each base, we have to define a different f(x) and find proper seeds.

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