((((((25210088873 + 80)3 + 12)3 + 450)3 + 894)3 + 3636)3 + 70756)3 + 97220

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:((((((25210088873 + 80)3 + 12)3 + 450)3 + 894)3 + 3636)3 + 70756)3 + 97220
Verification status (*):PRP
Official Comment (*):ECPP, Mills' prime
Unofficial Comments:This prime has 2 user comments below.
Proof-code(s): (*):FE1 : Morain, FastECPP
Decimal Digits:20562   (log10 is 20561.24316172)
Rank (*):73403 (digit rank is 1)
Entrance Rank (*):22561
Currently on list? (*):no
Submitted:6/5/2006 20:32:01 UTC
Last modified:3/11/2023 15:54:10 UTC
Database id:77907
Status Flags:Verify
Score (*):34.6719 (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.
Mills' prime (tolerated *)
Prime on list: no, rank 1
Subcategory: "Mills' prime"
(archival tag id 192157, tag last modified 2023-03-11 15:53:59)
Elliptic Curve Primality Proof (archivable *)
Prime on list: no, rank 93
Subcategory: "ECPP"
(archival tag id 192156, tag last modified 2024-04-24 05:37:25)

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 (11 Sep 2014):  (report abuse)
The number
is prime. Interested readers may read
for the origin of this number.

It has 20,562 decimal digits and the proof was built using fastECPP [1] on several networks of workstations. It was suggested as a challenge for primality proving. Since machines are more available than human time, letting them work for a somewhat unreasonnable amount of time is not an issue, as long as only one human check is needed from time to time. Thanks to stable power supply, and network, let alone a stable program, this record was possible. The computations were started on 32-bit machines (Sep-Oct 2005), and finished on nine 64-bit bi-processors (Feb-June 2006).

Cumulated timings are given w.r.t. AMD Opteron(tm) Processor 250 at 2.39 GHz.

1st phase: 1900 days (396 for sqrt; 384 for Cornacchia; 1353 for PRP tests)
2nd phase: 319 days (8 days for building all H_D's; 277 for solving H_D mod p)
The certificate (48Mb compressed) can be found at:
It took 10 days to check the 1765 proof steps on a single processor.

F. Morain

[1] http://www.lix.polytechnique.fr/Labo/Francois.Morain/Articles/fastecpp-final.ps.gz

Chris Caldwell writes (11 Sep 2014):  (report abuse)
The prime Franĉois Morain so masterfully proved prime, was found to be a PRP by Andrey Kulsha in Sept. 2002. For some reason I missed (or forgot) about that, and credited Phil Carmody for the find in
Phil did indeed independently (re)discover the PRP, but Andrey was first! But of course none of us could prove it prime!

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.
machineWinXP P4 2.2GHz By K
notesCommand: pfgw.exe -n -f -tc -q"(((((((2521008881+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220" 2>&1 PFGW Version 1.2.0 for Windows [FFT v23.8] Primality testing (((((((2521008881+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 6252599 Running N-1 test using base 3 Running N-1 test using base 5 Running N-1 test using base 11 Running N+1 test using discriminant 19, base 1+sqrt(19) Calling N-1 BLS with factored part 0.05% and helper 0.03% (0.17% proof) (((((((2521008881+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220 is Fermat and Lucas PRP! (351.7329s+0.0394s) [Elapsed time: 356 seconds]
modified2020-07-07 22:30:42
created2006-06-05 20:58:09

machineGenToo P3 400MHz
notesCommand: /home/caldwell/client/pfgw -o -f -q"((((((2521008887^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] ((((((2521008887^3+80)^3+.........^3+3636)^3+70756 1/1 trial factoring to 6252599 ((((((2521008887^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220 has no small factor. [Elapsed time: 10.931 seconds]
modified2020-07-07 22:30:42
created2006-07-07 15:39:24

Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.
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