2007-09-04, 07:10 | #1 |
Feb 2004
France
3^{2}×103 Posts |
30th Wagstaff prime
Hi,
The 30th Wagstaff number (OEIS A000978) 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: http://primes.utm.edu/top20/page.php?id=67 (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 ---------------------------------------------------------------------- The Official Announcement in NMBRTHRY@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...2737.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/NewMe...onjecture.html [2] Math. Comp. 76, 493--505. -- |
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