Hi,
A Wagstaff number is a number of the form: (2^q+1)/3, where q is a prime.
As a member of the DUR gang (where DUR means: Diepeveen, Underwood, and Reix),
which is a project initiated (about 18 months ago) and managed by Vincent
Diepeveen for finding new Wagstaff PRPs by means of the LLR tool built by Jean Penné
(based on the gwnum library by George Woltman) who implemented the Vrba-Reix PRP test
imagined by Anton Vrba based on previous work of mine about using LLT cycles for
Mersennes, I've found a new and big Wagstaff PRP.
Previous Wagstaff PRP was found by Vincent Diepeveen : (2^986191+1)/3 in June 2008.
This Thursday 18th of February 2010, after testing about 43000 Wagstaff candidates
during about 16 months, I've got the following message by Jean Penné's fresh
version 3.8.0 of LLR :
(2^4031399+1)/3 is Vrba-Reix PRP!
This Wagstaff number has 1,213,572 digits and it is the 3rd biggest PRP ever found.
See:
http://www.primenumbers.net/prptop/prptop.php?page=1#haut.
I've done a second verification on a Nehalem core, still with LLR 3.8.0 : OK after
less than 3 hours.
I've also ran the PFGW tool (with -t option) on Nehalem, which said: (2^4031399+1)/3
is PRP! after about 15 hours. See below for details.
I guess that another independent verification is required before this PRP is accepted.
However, testing with two different methods is probably enough.
I'm very happy !
I thank a lot Vincent for proposing me to take part in this project. He managed the
exponents ranges and he sieved all the exponents I've tested. He and Paul Underwood
tested many exponents too. But I think I deserved to find this PRP since I've tested
a very big part of all candidate exponents, and also because the Vrba-Test used by
LLR 3.8.0 was imagined after my research about Mersennes and LLT (Lucas-Lehmer Test)
properties.
For those interested by the theory behind the Vrba-Reix test, have a look at:
http://trex58.wordpress.com/math2matiques/ .
People having ideas to prove the 3 conjectures are welcome !!
And there is a 100€ reward !
I dedicate this Wagstaff PRP to my wife, Catherine, who died the 23th of October, 2006,
and who never understood why I was spending so much time about these crazy and unuseful
Number Theory mathematic. Maybe I should have spent more time with her than I spent as
an amateur with Mersenne and LLT theory...
All this long email is aimed to push Number Theory guys to look for a proof for the 3
conjectures my-self, Vrba and Gerbicz made about 2 years ago.
Regards,
Tony Reix
$./pfgw -t -q"(2^4031399+1)/3"
PFGW Version 20091218.x86_Dev (Beta 'caveat utilitor') [GWNUM 25.13]
Primality testing (2^4031399+1)/3 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^4031399+1)/3 is PRP! (56473.2345s+1.3909s)