20100224, 21:43  #23 
Feb 2004
France
3^{2}·103 Posts 
Nice !

20100224, 22:09  #24 
Jan 2007
Germany
2·239 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. 
20100224, 22:18  #25  
Feb 2004
France
1637_{8} Posts 
Quote:
Continue to search. If you find a universal (for all Wagstaff primes) seed that leads to a cycle of la ength that divides q1, I'm very interested !! Tony 

20100224, 22:40  #26 
Bemusing Prompter
"Danny"
Dec 2002
California
2^{5}×7×11 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 emailed MathWorld and Chris Caldwell but never got any response. Anyone else want to try?

20100225, 04:32  #27 
Jan 2007
Germany
2·239 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 20100225 at 04:37 
20100225, 11:06  #28  
Feb 2004
France
3^{2}×103 Posts 
Ned help...
Quote:
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^22 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 N1 (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 q2. 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 statusunknown candidate). Moreover, this technique (use Cycles or Trees) can be extended to other kinds of numbers and with other functions than x^22. 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. Look at this page for all explainations and useful paper. Tony 

20100225, 13:08  #29 
"Phil"
Sep 2002
Tracktown, U.S.A.
2137_{8} Posts 
Your number now is listed on the Lifchitz site:
http://www.primenumbers.net/prptop/prptop.php The third megadigit prp  congratulations, and go find another! 
20100225, 20:01  #30  
Feb 2004
France
3^{2}·103 Posts 
Quote:
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 

20100225, 20:31  #31 
Jun 2003
14EF_{16} Posts 
At the risk of derailing the thread, the VR test can be generalized for nonbase2 primes (both for generalized repunits (b^n1)/(b1) and generalized wagstaff (b^n+1)/(b+1))

20100225, 20:36  #32 
Feb 2004
France
3^{2}×103 Posts 

20100226, 01:25  #33 
Jun 2003
23·233 Posts 
For base 3:
Define f(x) = x^33*x Test for N=(3^p1)/(31): s_{0}=4, s_{i+1}=f(s_{i}), s_{p}s_{1} (mod N) Test for N=(3^p+1)/(3+1): s_{0}=8, s_{i+1}=f(s_{i}), s_{p}s_{1} (mod N) For each base, we have to define a different f(x) and find proper seeds. 
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