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: ((((((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 20561.2431617196) Rank (*): 54276 (digit rank is 1) Entrance Rank (*): 22561 Currently on list? (*): short Submitted: 6/5/2006 15:32:01 CDT Last modified: 7/7/2006 10:39:35 CDT Database id: 77907 Status Flags: Verify Score (*): 34.6719 (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.

Mills' prime (tolerated *)

Prime on list: no, rank 1
Subcategory: "Mills' prime"
(archival tag id 192157, tag last modified 2009-02-12 20:01:29)

Elliptic Curve Primality Proof (archivable *)

Prime on list: yes, rank 3
Subcategory: "ECPP"
(archival tag id 192156, tag last modified 2012-02-19 19:20:39)

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 (9 Aug 2007): 

The number (((((((((23+3)3+30)3+6)3+80)3+12)3+450)3+894)3+3636)3+70756)3+97220 is prime. Interested readers may read http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html 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: http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/Certif/mills2.certif.gz 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 (9 Aug 2007): 

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 http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html 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.

field value prime_id 77907 person_id 9 machine WinXP P4 2.2GHz By K what prp notes Command: 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] modified 2006-07-14 06:27:04 created 2006-06-05 15:58:09 id 84439

field value prime_id 77907 person_id 9 machine GenToo P3 400MHz what trial_divided notes Command: /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] modified 2006-07-14 06:27:03 created 2006-07-07 10:39:24 id 85237