(2⁴²⁷³⁷ + 1)/3

This prime's information:

Archival tags:

There are certain forms classed as archivable: these prime may (at times) remain on this list even if they do not make the Top 5000 proper.  Such primes are tracked with archival tags.

Elliptic Curve Primality Proof (archivable *)

Prime on list: no, rank 238
Subcategory: "ECPP"
(archival tag id 193533, tag last modified 2024-10-27 10:37:10)

Generalized Lucas Number (archivable *)

Prime on list: no, rank 39
Subcategory: "Generalized Lucas Number"
(archival tag id 193532, tag last modified 2023-10-24 02:37:13)

Wagstaff (archivable *)

Prime on list: yes, rank 6
Subcategory: "Wagstaff"
(archival tag id 193534, tag last modified 2023-10-24 02:37:14)

User comments about this prime (disclaimer):

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François Morain writes (11 Sep 2014):  (report abuse)

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. ---------------- [1] http://t5k.org/mersenne/NewMersenneConjecture.html [2] Math. Comp. 76, 493--505.

Verification data:

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