At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly. This list is the most important databases at The Prime Pages: 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:

field (help) value Description: (242737 + 1)/3 Verification status (*): PRP Official Comment: ECPP, generalized Lucas number, Wagstaff Unofficial Comments: This prime has 1 user comment below. Proof-code(s): (*): M : Morain Decimal Digits: 12865   (log10 is 12864.641803437) Rank (*): 61005 (digit rank is 1) Entrance Rank (*): 32933 Currently on list? (*): short Submitted: 8/28/2007 13:19:44 CDT Last modified: 8/28/2007 13:27:29 CDT Database id: 82071 Status Flags: Verify Score (*): 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: yes, rank 19
Subcategory: "ECPP"
(archival tag id 193533, tag last modified 2013-05-18 19:50:29)

Generalized Lucas Number (archivable *)

Prime on list: no, rank 21
Subcategory: "Generalized Lucas Number"
(archival tag id 193532, tag last modified 2013-01-10 05:50:32)

Wagstaff (archivable *)

Prime on list: yes, rank 1
Subcategory: "Wagstaff"
(archival tag id 193534, tag last modified 2009-03-17 13:20:28)

User comments about this prime (disclaimer):

User comments are allowed to convey mathematical information about this number, how it was proven prime.... See our guidelines and restrictions.

François Morain writes (28 Aug 2007): 

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.

field value prime_id 82071 person_id 9 machine RedHat P4 P4 what prp notes Command: /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] modified 2007-09-12 06:50:18 created 2007-08-28 13:20:30 id 93262

field value prime_id 82071 person_id 9 machine RedHat P4 P4 what trial_divided notes Command: /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] modified 2007-09-12 06:50:18 created 2007-08-28 13:22:01 id 93263