20211126, 22:14  #1 
Feb 2004
France
3×311 Posts 
Universal seeds of the LLT for Mersenne and Wagstaff numbers
You probably know that there are 3 Universal Seeds for starting the LLT for Mersenne numbers: 4, 10, and 2/3.
It appears that 2/3 modulo Mq = (2^q+1)/3 = Wq, a Wagstaff number. Also, you may know that there is a conjecture about proving that a Wagstaff number Wq is prime by using a cycle of the DiGraph under x^22 modulo Wq. The seed we found years ago is : 1/4. And kijinSeija found that 1/4 = W(q2). Moreover, I've found that 1/10 and 1/Mq seem also to be usable as Universal Seed for this conjecture, building a new bridge between Mersenne and Wagstaff primes, since the seeds of the Wagstaff conjecture are inverse modulo Wq of the Universal Seeds of the Mersenne LLT. Pari/gp: T(q)={M=2^q1;W=(2^q+1)/3;S0=Mod(1/4,W);S=S0;print(S);for(i=1,q+1,S=Mod(S^22,W);print(S))} Try and replace 1/4 by 1/10 and 1/Mq. It's perfectly clear for 1/4 and 1/10. It's a little bit different for 1/Mq. It seems to depend if q=4k+1 or 4k1. It needs deeper study. 
20211126, 22:38  #2  
Sep 2002
Database er0rr
5×29×31 Posts 
Quote:
Code:
wag(q)=W=(2^q+1)/3;S0=S=Mod(3/2,W);for(i=2,q,S=S^22);S==S0; Are 4, 10 and 2/3 the only ones for Mersenne? I remember Lehmer had said something about these. Last fiddled with by paulunderwood on 20211126 at 22:48 

20211126, 23:49  #3 
Sep 2002
Database er0rr
10617_{8} Posts 
More on the seed 3/2 for Wagstaff numbers.
Mod(Mod(x,W),x^23/2*x+1) is at the heart because S = S0 = 3/2 = x+1/x which leads to the recurrence S=S^22 mod W The solution for x is ( 3/2 + sqrt( (3/2)^24 ) / 2 = ( 3 + 2 * sqrt(2) ) / 4 That is 4*x  3 == + 2 * sqrt(2) So we can use a power of Mod(Mod(4*x3,W),x^28). By trial and error Mod(Mod(4*x3,W),x^28)^(W+1)+119 == 0 Edit. That marked red is wrong, but somehow the result is good! Stumped! Last fiddled with by paulunderwood on 20211127 at 01:30 
20211127, 00:35  #4  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·29·127 Posts 
Quote:
The ninth page of Bas Jensen's PhD number theory thesis https://scholarlypublications.univer...3A2919366/view makes reference to a 1996 conjecture by G. Woltman, proven 4 years later. Last fiddled with by kriesel on 20211127 at 00:37 

20211127, 05:08  #5  
Einyen
Dec 2003
Denmark
2×3^{2}×191 Posts 
Quote:
https://www.mersenneforum.org/showpo...0&postcount=46 A_{0}=A_{1}=4, A_{N}=14*A_{N1}A_{N2} B_{0}=B_{1}=10, B_{N}=98*B_{N1}B_{N2} Last fiddled with by ATH on 20211127 at 05:11 

20211127, 08:52  #6  
Feb 2004
France
3·311 Posts 
Quote:
I see that 2/3 = Wq was already known. 

20211127, 08:54  #7 
Feb 2004
France
3·311 Posts 

20211127, 09:22  #8 
Sep 2002
Database er0rr
5×29×31 Posts 
It is not so great since (3/2)^22 = 1/4 which is already known.
Just as (3)^((Mp  1)/2) == 1 mod Mp for Mersenne primes, we have (7)^((Wq  1)/2) == 1 mod Wq for Wagstaff PRPs. The latter can be 7^((Wq  1 )/2) == 1 mod Wq. Last fiddled with by paulunderwood on 20211127 at 10:09 
20211127, 13:35  #9 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·29·127 Posts 

20211130, 22:15  #10 
Feb 2004
France
3×311 Posts 
Hello.
I've spent some time searching for Universal Seeds for the LLTlike test for Wagstaff numbers. Here attached is a first set of findings. Everything has been checked for all known Wagstaff primes and for Wagstaff notprimes up to q = 10,000. It may be strange, but 23/8 mod Wq is a Universal seed! And 1/10 is, sometimes... Enjoy Tony 
20211208, 01:15  #11 
Bemusing Prompter
"Danny"
Dec 2002
California
5×499 Posts 
Probably a stupid question, but my knowledge is not up to scratch.
It is my understanding that "2/3" is shorthand for (2 mod M(p))(3 mod M(p))^{1} = W(p). I was indeed able to confirm that W(p) works as a seed value when testing whether M(p) is prime. However, it's not clear how (2 mod M(p))(3 mod M(p))^{1} evaluates to a Wagstaff number. I can see where the "2/3" comes from because 2 mod M(p) = 2 and 3 mod M(p) = 3 for p > 2. But it doesn't make sense to use a fraction as a seed value. Or does (3 mod M(p))^{1} mean something other than 1 / (3 mod M(p)) in this case? 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Status of Wagstaff testing? and testing Mersenne primes for Wagstaffness  GP2  Wagstaff PRP Search  414  20201227 08:11 
(New ?) Wagstaff/Mersenne related property  T.Rex  Wagstaff PRP Search  6  20191123 22:46 
P1 bounds calculation for Wagstaff numbers  axn  Software  10  20180724 06:27 
PRIMALITY PROOF for Wagstaff numbers!  AntonVrba  Math  96  20090225 10:37 
Bernoulli and Euler numbers (Sam Wagstaff project)  fivemack  Factoring  4  20080224 00:39 