(242737 + 1)/3

At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly.  This list is the most important PrimePages database: a collection of research, records and results all about prime numbers. This page summarizes our information about one of these primes.

This prime's information:

Description: (242737 + 1)/3 PRP ECPP, generalized Lucas number, Wagstaff This prime has 1 user comment below. M : Morain 12865   (log10 is 12864.641803437) 77651 (digit rank is 1) 32933 short 8/28/2007 18:19:44 UTC 3/11/2023 15:54:10 UTC 82071 Verify 33.2205 (normalized score 0)

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 190
Subcategory: "ECPP"
Generalized Lucas Number (archivable *)
Prime on list: no, rank 38
Subcategory: "Generalized Lucas Number"
Wagstaff (archivable *)
Prime on list: yes, rank 5
Subcategory: "Wagstaff"

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://primes.utm.edu/mersenne/NewMersenneConjecture.html [2] Math. Comp. 76, 493--505.

Verification data:

The Top 5000 Primes is a list for proven primes only. In order to maintain the integrity of this list, we seek to verify the primality of all submissions.  We are currently unable to check all proofs (ECPP, KP, ...), but we will at least trial divide and PRP check every entry before it is included in the list.
fieldvalue
prime_id82071
person_id9
machineRedHat P4 P4
whatprp
notesCommand: /home/caldwell/client/pfgw -tc -q"(2^42737+1)/3" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2^42737+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Running N-1 test using base 7 Running N-1 test using base 13 Running N-1 test using base 19 Running N+1 test using discriminant 29, base 2+sqrt(29) Running N+1 test using discriminant 29, base 3+sqrt(29) Running N+1 test using discriminant 29, base 5+sqrt(29) Running N+1 test using discriminant 29, base 6+sqrt(29) Running N+1 test using discriminant 29, base 8+sqrt(29) Running N+1 test using discriminant 29, base 9+sqrt(29) Calling N+1 BLS with factored part 0.98% and helper 0.15% (3.08% proof) (2^42737+1)/3 is Fermat and Lucas PRP! (414.8100s+0.0000s) [Elapsed time: 6.98333333333333 minutes]
modified2020-07-07 22:30:40
created2007-08-28 18:20:30
id93262

fieldvalue
prime_id82071
person_id9
machineRedHat P4 P4
whattrial_divided
notesCommand: /home/caldwell/client/pfgw -o -f -q"(2^42737+1)/3" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] trial factoring to 3771838 (2^42737+1)/3 has no small factor. [Elapsed time: 4.875 seconds]
modified2020-07-07 22:30:40
created2007-08-28 18:22:01
id93263

Query times: 0.0003 seconds to select prime, 0.0005 seconds to seek comments.