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: (163393613-1)/16338 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: 15219   (log10 is 15218.170434239) Rank (*): 38432 (digit rank is 2) Entrance Rank (*): 32218 Currently on list? (*): short Submitted: 3/8/2008 06:21:15 CDT Last modified: 3/8/2008 06:50:18 CDT Database id: 84344 Status Flags: Verify Score (*): 33.7408 (normalized score 0.001)

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 16
Subcategory: "Generalized Repunit"
(archival tag id 192506, 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 (8 Mar 2008): 

Certificate at http://www.primes.viner-steward.org/andy/X/3FD30E1D.html

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 84344 person_id 9 machine RedHat P4 P4 what trial_divided notes Command: /home/caldwell/client/pfgw -o -f -q"(16339^3613-1)/16338" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] trial factoring to 4521313 (16339^3613-1)/16338 has no small factor. [Ellapsed time: 6.013 seconds] modified 2008-04-18 10:43:09 created 2008-03-08 06:22:09 id 97817

field value prime_id 84344 person_id 9 machine RedHat P4 P4 what prp notes Command: /home/caldwell/client/pfgw -tc -q"(16339^3613-1)/16338" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (16339^3613-1)/16338 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 11 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) 101116 bit request FFT size=(6144,17) Running N+1 test using discriminant 17, base 14+sqrt(17) 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) 101124 bit request FFT size=(6144,17) Calling N-1 BLS with factored part 0.73% and helper 0.02% (2.20% proof) (16339^3613-1)/16338 is Fermat and Lucas PRP! (110.8600s+0.0000s) [Elapsed time: 1.85 minutes] modified 2008-04-18 10:43:09 created 2008-03-08 06:25:34 id 97820