(V(77786, 1, 6453) + 1)/(V(77786, 1, 27) + 1)
(Another of the Prime Pages' resources)
The Largest Known Primes Icon
  View this page in:   language help

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(77786, 1, 6453) + 1)/(V(77786, 1, 27) + 1)
Verification status (*):PRP
Official Comment:Lehmer primitive part
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):x25 : Water, Broadhurst, Primo, OpenPFGW
Decimal Digits:31429   (log10 is 31428.932648348)
Rank (*):59330 (digit rank is 1)
Entrance Rank (*):47767
Currently on list? (*):short
Submitted:12/26/2012 11:13:59 CDT
Last modified:12/26/2012 11:50:25 CDT
Database id:110481
Status Flags:Verify
Score (*):35.9835 (normalized score 0.0003)

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 1
Subcategory: "Lehmer primitive part"
(archival tag id 214827, tag last modified 2012-12-26 11:50:27)

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 (11 Sep 2014): 

CHG proof at 31429 digits and 26.19% factorization with details in http://physics.open.ac.uk/~dbroadhu/cert/lh77786.zip and a certificate compressed to 25MB in http://physics.open.ac.uk/~dbroadhu/cert/lh77786_cert.gp.gz

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.
machineRedHat P4 P4
notesCommand: /home/caldwell/client/pfgw -o -f -q"(lucasV(77786,1,6453)+1)/(lucasV(77786,1,27)+1)" 2>&1
PFGW Version [GWNUM 26.5]
(lucasV(77786,1,....V(77786,1,27)+1) 1/1 mro=0

trial factoring to 9867605
(lucasV(77786,1,6453)+1)/(lucasV(77786,1,27)+1) has no small factor.
[Elapsed time: 25.563 seconds]
modified2013-02-02 20:51:49
created2012-12-26 11:18:01

machineRedHat P4 P4
notesCommand: /home/caldwell/client/pfgw -tc -q"(lucasV(77786,1,6453)+1)/(lucasV(77786,1,27)+1)" 2>&1
PFGW Version [GWNUM 26.5]
Primality testing (lucasV(77786,1,6453)+1)/(lucasV(77786,1,27)+1) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 59
Running N-1 test using base 109
Running N-1 test using base 113
Running N-1 test using base 151
Running N+1 test using discriminant 163, base 24+sqrt(163)
Calling N-1 BLS with factored part 1.58% and helper 0.06% (4.80% proof)
(lucasV(77786,1,6453)+1)/(lucasV(77786,1,27)+1) is Fermat and Lucas PRP! (819.2734s+0.0062s)
[Elapsed time: 13.65 minutes]
modified2013-02-02 20:51:49
created2012-12-26 11:23:01

Query times: 0.0004 seconds to select prime, 0.0006 seconds to seek comments.