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: (27284231-1)/2727 Verification status (*): PRP Official Comment: Generalized repunit Unofficial Comments: This prime has 1 user comment below. Proof-code(s): (*): c13 : Steward, OpenPFGW, Primo Decimal Digits: 14534   (log10 is 14533.621827342) Rank (*): 39545 (digit rank is 1) Entrance Rank (*): 29242 Currently on list? (*): short Submitted: 4/21/2007 12:42:36 CDT Last modified: 4/21/2007 13:02:49 CDT Database id: 80117 Status Flags: Verify Score (*): 33.5983 (normalized score 0.0008)

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.

Generalized Repunit (archivable *)

Prime on list: yes, rank 20
Subcategory: "Generalized Repunit"
(archival tag id 192567, tag last modified 2009-07-12 03:50:27)

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.

Andrew A. D. Steward writes (21 Apr 2007): 

Certificate at http://www.primes.viner-steward.org/andy/H/0AA81087.html (Originally proved 2005-12-10)

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 80117 person_id 9 machine RedHat P4 P4 what trial_divided notes Command: /home/caldwell/client/pfgw -o -f -q"(2728^4231-1)/2727" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] trial factoring to 4302379 (2728^4231-1)/2727 has no small factor. [Elapsed time: 5.480 seconds] modified 2007-04-21 22:36:36 created 2007-04-21 12:52:23 id 89340

field value prime_id 80117 person_id 9 machine RedHat P4 P4 what prp notes Command: /home/caldwell/client/pfgw -tc -q"(2728^4231-1)/2727" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2728^4231-1)/2727 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(6144,21) to FFT(6144,20) Reduced from FFT(6144,20) to FFT(6144,19) Reduced from FFT(6144,19) to FFT(6144,18) Reduced from FFT(6144,18) to FFT(6144,17) Reduced from FFT(6144,17) to FFT(6144,16) 96568 bit request FFT size=(6144,16) Running N-1 test using base 47 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(6144,21) to FFT(6144,20) Reduced from FFT(6144,20) to FFT(6144,19) Reduced from FFT(6144,19) to FFT(6144,18) Reduced from FFT(6144,18) to FFT(6144,17) Reduced from FFT(6144,17) to FFT(6144,16) 96568 bit request FFT size=(6144,16) Running N-1 test using base 53 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(6144,21) to FFT(6144,20) Reduced from FFT(6144,20) to FFT(6144,19) Reduced from FFT(6144,19) to FFT(6144,18) Reduced from FFT(6144,18) to FFT(6144,17) Reduced from FFT(6144,17) to FFT(6144,16) 96568 bit request FFT size=(6144,16) Running N-1 test using base 59 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(6144,21) to FFT(6144,20) Reduced from FFT(6144,20) to FFT(6144,19) Reduced from FFT(6144,19) to FFT(6144,18) Reduced from FFT(6144,18) to FFT(6144,17) Reduced from FFT(6144,17) to FFT(6144,16) 96568 bit request FFT size=(6144,16) Running N+1 test using discriminant 67, base 18+sqrt(67) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(6144,21) to FFT(6144,20) Reduced from FFT(6144,20) to FFT(6144,19) Reduced from FFT(6144,19) to FFT(6144,18) Reduced from FFT(6144,18) to FFT(6144,17) Reduced from FFT(6144,17) to FFT(6144,16) 96576 bit request FFT size=(6144,16) Calling N-1 BLS with factored part 0.69% and helper 0.09% (2.18% proof) (2728^4231-1)/2727 is Fermat and Lucas PRP! (174.0200s+0.0000s) [Elapsed time: 174 seconds] modified 2007-04-21 22:36:36 created 2007-04-21 12:59:55 id 89346