# [Fwd: (2^42737+1)/3 is prime] 12,865 digits (OEIS A000978)

tony.reix at laposte.net tony.reix at laposte.net
Mon Sep 3 15:04:59 CEST 2007

```Hi,

The 30th Wagstaff number (OEIS A000978) has been proved prime by François Morain, thanks to FastECPP.
Thus, some comments should be changed in:
http://www.research.att.com/~njas/sequences/A000978

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 Reix

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The Official Announcement in NMBRTHRY at LISTSERV.NODAK.EDU :

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:

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

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] http://primes.utm.edu/mersenne/NewMersenneConjecture.html

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

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