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: (V(8162, 1, 2307)-1)/(V(8162, 1, 3)-1) Verification status (*): Proven Official Comment: Lehmer primitive part Unofficial Comments: This prime has 2 user comments below. Proof-code(s): (*): x25 : Water, Broadhurst, Primo, OpenPFGW Decimal Digits: 9013   (log10 is 9012.77932935) Rank (*): 45230 (digit rank is 1) Entrance Rank (*): 31548 Currently on list? (*): short Submitted: 12/28/2005 22:10:41 CDT Last modified: 12/29/2005 23:05:56 CDT Database id: 76586 Status Flags: none Score (*): 32.1179 (normalized score 0.0002)

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.

Lehmer primitive part (archivable *)

Prime on list: yes, rank 20
Subcategory: "Lehmer primitive part"
(archival tag id 194414, tag last modified 2009-05-10 04:50:23)

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.

David Broadhurst writes (28 Dec 2005): 

A BLS proof was obtained in November 2003, using the factorization percentage recorded in http://groups.yahoo.com/group/primeform/files/GenLuc/primlehm.txt at that time. A certificate will be linked to a subsequent comment.

David Broadhurst writes (29 Dec 2005): 

BLS proof with helpers in http://physics.open.ac.uk/~dbroadhu/cert/b6921d.zip

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 76586 person_id 9 machine WinXP Pentium M 1.7GHz what prime notes Command: pfgw.exe -n -f -tc -hp_76586.txt -q"(lucasV(8162,1,2307)-1)/(lucasV(8162,1,3)-1)" 2>&1 PFGW Version 20030811.Win_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Reading factors from helper file p_76586.txt trial factoring to 2567875 Running N-1 test using base 19 Primality testing (lucasV(8162,1,2307)-1)/(lucasV(8162,1,3)-1) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 29 Calling N-1 BLS with factored part 35.53% and helper 0.14% (106.74% proof) (lucasV(8162,1,2307)-1)/(lucasV(8162,1,3)-1) is prime! (170.1879s+0.3195s) Running N+1 test using discriminant 103, base 4+sqrt(103) [Elapsed time: 172 seconds] modified 2005-12-30 08:18:46 created 2005-12-29 23:02:55 id 82116

field value prime_id 76586 person_id 9 machine Linux P4 2.8GHz what trial_divided notes PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] 6016298106...9679851310.........1120081663380116 1/1 trial factoring to 2567875 6016298106...6926054401 has no small factor. [Elapsed time: 15.590 seconds] modified 2005-12-29 09:10:59 created 2005-12-28 22:22:20 id 82071