30th Wagstaff prime
 

Hi,

The 30th Wagstaff number (OEIS A000978) [B][tex]\frac{2^{42737}+1}{3}[/tex][/B] has been proved prime by François Morain, thanks to FastECPP.

Read the official announcement here below.
Remember that the 9 remaining probable Wagstaff primes have been found by Renaud Lifchitz, thanks to a modified version of prime95.

New record already appears in primes.edu: [URL="http://primes.utm.edu/top20/page.php?id=67"]http://primes.utm.edu/top20/page.php?id=67[/URL]

(Some comments should be changed in: [url]http://www.research.att.com/~njas/sequences/A000978[/url] )


Notice that there is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT):

Let p be a prime integer > 3 , and Np = 2^p+1 and Wp = N/3 .
S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod Np) ;
Wp is prime iff S(p-1) == S(0) (mod Wp) .

So, it should be possible to reuse prime95 for proving very quickly that this kind of number is prime or not (once a proof has been found, for sure !!!).
Or one could find a new (very) probable Wagstaff prime.

Regards,

Tony


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The Official Announcement in [email]NMBRTHRY@LISTSERV.NODAK.EDU[/email] :


The number N = (2^42737+1)/3 is prime.

It is related to the conjecture of Bateman, Selfridge and Wagstaff, see [1]. Previous exponents p leading to prime values of N_p = (2^p+1)/3 can also be found at [1]. The next value of p for which N_p is a probable prime is p=83339, which might not be undoable in a near future.

The number N has 12,865 decimal digits and the proof was built using fastECPP [2] on several networks of workstations.

Cumulated timings are given w.r.t. AMD Opteron(tm) Processor 250 at 2.39 GHz.

1st phase: 218 days (72 for sqrt; 8 for Cornacchia; 134 for PRP tests) 2nd phase: 93 days (2 days for building all H_D's; 83 for solving H_D mod p)

The certificate (>19Mb compressed) can be found at:

[url]http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/Certif/bsw42737.certif.gz[/url]

It took 2 days to check the 1165 proof steps on a single processor.

Acknowledgment: thanks to Tony Reix for having pushed me to come back to the primality of these numbers.

F. Morain

[1] [url]http://primes.utm.edu/mersenne/NewMersenneConjecture.html[/url]

[2] Math. Comp. 76, 493--505.

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