20100226, 05:00  #34  
Feb 2004
France
2·3^{3}·17 Posts 
GVR !!
Quote:
Look at HC. Williams' book ("Edouard Lucas and Primality testing"), pages 77 and 78. The Lucas sequences naturally leads to Chebyshev polynomials : Sn and Cn, the same as Un and Vn for Lucas sequences. Z(n)=x*Z(n1)Z(n2) . So, as an example, with b=5, and with the following gp/PARI program: Code:
G(q)=W=(5^q+1)/6;print(q," ",isprime(W));S0=2^5;S=S0;for(i=1,q,S=Mod(S^55*S^3+5*S,W); if(i==1,S1=S));if(S==S1,print("prime")) forprime(j=3,100,G(j)) However, it missed: q=3,229,347 !! Probably due to a bad seed.... Though, for W=(3^q+1)/4, your seed seems perfect. W is prime for q=3,5,7,13,23,43,281,359,487,577 So, I've provided a general f(x) for each base of Generalized Wagstaff family. So now, if we find a formula for the seed, we have a GVR PRP test for Generalized Wagstaff numbers !! Yes, sure, there is a RICH area of research there... Tony 

20100226, 06:30  #35 
Bemusing Prompter
"Danny"
Dec 2002
California
2,411 Posts 

20100226, 14:44  #36 
Jun 2005
2·7^{2} Posts 
My congratulations to Tony, may he soon top the Lifchitz pages.
Tony your long patience has been rewarded! Is there a huge gap between 986191 4031399 or have all primes not been tested yet between the the known largest wagstaff prps? 
20100226, 17:01  #37  
Feb 2004
France
2×3^{3}×17 Posts 
Huge gap
Quote:
Quote:
Tony 

20100227, 02:06  #38  
Jun 2003
19×271 Posts 
I rediscovered these in my investigations
Quote:
I am close to working out how these tests are working (looks like glorified fermat test) 

20100227, 03:00  #39  
"Phil"
Sep 2002
Tracktown, U.S.A.
10001011111_{2} Posts 
Quote:
Phil 

20100227, 08:18  #40  
Feb 2004
France
1110010110_{2} Posts 
Quote:
However, I guess that it is also possible to build a PRP test for base b generalized Wagstaff numbers with x^22 .... T. 

20100227, 08:31  #41  
Feb 2004
France
2·3^{3}·17 Posts 
Great idea !
Quote:
Nice proposal ! Good idea ! I'll talk with Vincent (Diepeeven), who initiated this project and is managing it. I think he will be very happy to have his idea showing to be a very smart one and to attract more people. It depends if he has time, too. He's involved in his Diep chess program and he, from time to time, is very busy with it. On my side, I am interested by many other subjects (my job..., philosophy, my Blog, friends, poetry, readings, athéism, my children, women and sex ! :) ), so I do not plan to spend on Number Theory and PRP search as much time as I spent in the past. However, I plan to continue on this as long as I can. So much fun ! So, I would be very happy that a bigger Wagstaff PRP is found ! even by someone else. That would attract more people on looking at these damned Conjectures that Vrba, Gerbicz and myself built 2 years ago now. We need a proof ! And, with a proof for VrbaReix conjecture, I would have found a 1.2 M digits prime !! Tony 

20100227, 23:43  #42  
Sep 2006
The Netherlands
769 Posts 
Industry grade primes forum
Quote:
It is true we profit from great work that has been done for the GIMPS search so far, as everything that applies to Mersenne also applies to Wagstaff. I don't see how other formula's can rival with it in the long run in terms of PRP finding. Of course i didn't study math, but we just need to test, just like Mersenne, exponent numbers that are prime themselves. 60% gets trial factored. So after we have determined that the most useless form of wasting CPU power is the search for the holy infinite sized grail as we try here, then searching for Wagstaff from all industry grade primes is the most efficient one i'd argue. Vincent 

20100227, 23:59  #43  
Sep 2006
The Netherlands
301_{16} Posts 
Quote:
However odds are relative small. Yet there seems bug in core2 which avoided it from functioning very well and there is always a possibility for other problems or misses. So we'll double check up to the largest Wagstaff found to date; i do not know our odds of finding one there. the bigger ranges more chances maybe. But maybe one is real closeby who knows. You never know. This lottery is better than joining other lotteries i'd argue. Vincent 

20100303, 00:48  #44  
Feb 2005
374_{8} Posts 
Quote:
Indeed, if for odd prime , then , while the multiplicative order of 2 modulo is . Clearly, , implying that . Furthermore, , implying that . Therefore, always passes Miller–Rabin primality test base 2 (even if is not prime). 

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