67535122 + 51226753
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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:67535122 + 51226753
Verification status (*):PRP
Official Comment:ECPP
Unofficial Comments:This prime has 2 user comments below.
Proof-code(s): (*):FE1 : Morain, FastECPP
Decimal Digits:25050   (log10 is 25049.845444116)
Rank (*):64449 (digit rank is 1)
Entrance Rank (*):37494
Currently on list? (*):short
Last modified:2/19/2012 18:50:29 CDT
Database id:104822
Status Flags:Verify
Score (*):35.2824 (normalized score 0.0001)

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 11
Subcategory: "ECPP"
(archival tag id 214047, tag last modified 2020-02-10 21:20:06)

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.

Fran├žois Morain writes (11 Sep 2014): 
I am glad to announce that the number 6753^5122+5122^6753 is prime (a number taken from Paul Leyland's tables, see http://www.leyland.vispa.com/numth/primes/xyyx.htm). It has 25050 decimal digits and was proven prime using fastECPP.

Calendar time was 2010/09/01 -- 2010/10/15. Computations were done on a network of bi-core i7 quad-core using GMP-5.0.2, mpfr, mpc, mpfrcx, open MPI. Computing the DOWNRUN sequence took 1696 days, and the proving part 282 days (among which only 5 days for computing the class polynomials).

The gzipped certificate of the 2024 steps needs 55 Mb. Checking this certificate on a single machine takes 8 days. The certificate is available at:


Chris Caldwell writes (11 Sep 2014): 
This number was found to be a PRP by Anatoly Selevich according to Paul Leyland's tables.

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.
machineDitto P4 P4
notesCommand: /home/ditto/client/pfgw -tc -q"6753^5122+5122^6753" 2>&1
PFGW Version [GWNUM 26.5]
Primality testing 6753^5122+5122^6753 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 37, base 14+sqrt(37)
Calling N-1 BLS with factored part 0.04% and helper 0.00% (0.13% proof)
6753^5122+5122^6753 is Fermat and Lucas PRP! (239.2453s+0.0015s)
[Elapsed time: 3.98 minutes]
modified2012-02-19 19:07:47
created2012-02-19 18:33:20

machineDitto P4 P4
notesCommand: /home/ditto/client/pfgw -o -f -q"6753^5122+5122^6753" 2>&1
PFGW Version [GWNUM 26.5]
6753^5122+5122^6753 1/1 mro=0

trial factoring to 7732622
6753^5122+5122^6753 has no small factor.
[Elapsed time: 21.454 seconds]
modified2012-02-19 19:07:47
created2012-02-19 18:35:02

Query times: 0.0004 seconds to select prime, 0.0007 seconds to seek comments.