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.