(2963³⁸²¹ - 1)/2962

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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 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:	(2963³⁸²¹ - 1)/2962
Verification status (*):	PRP
Official Comment (*):	Generalized repunit
Unofficial Comments:	This prime has 1 user comment below.
Proof-code(s): (*):	CH2 : Wu_T, OpenPFGW, Primo, CHG
Decimal Digits:	13263 (log₁₀ is 13262.015055251)
Rank (*):	81978 (digit rank is 2)
Entrance Rank (*):	26396
Currently on list? (*):	no
Submitted:	9/19/2005 12:06:45 UTC
Last modified:	3/11/2023 15:54:10 UTC
Database id:	75630
Status Flags:	Verify
Score (*):	33.3148 (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.

Generalized Repunit (archivable *)
Prime on list: no, rank 68
Subcategory: "Generalized Repunit"
(archival tag id 193236, tag last modified 2025-01-20 23:37:21)

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.
Chris Caldwell writes (11 Sep 2014): (report abuse)
David Broadhurst writes: The certificate and the validator are in
http://physics.open.ac.uk/~dbroadhu/cert/ch2_cert.zip (3.3MB)
If you unzip that file and run
Phi_3821_2963_cert.gp
using GP 2.2.10 or 2.2.11, you should find that it is validated rather quickly:
> Testing a PRP called "Phi_3821_2963".
> LLL[1, 1] with [h, u]=[5, 1] has
> norm/bound=4.436400008 E-688 and witness=5.
...
> LLL[46, 1] with [h, u]=[16, 7] has
> norm/bound=7.41270031 E-16 and witness=41.
> Validated in 35 sec.

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 75630

person_id 9

machine Linux P4 2.8GHz

what prp

notes Command: /home/caldwell/client/pfgw -f -tc -q"(2963^3821-1)/2962" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2963^3821-1)/2962 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 3897723 Running N-1 test using base 7 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) 88120 bit request FFT size=(6144,16) Running N-1 test using base 23 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) 88120 bit request FFT size=(6144,16) Running N-1 test using base 41 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) 88120 bit request FFT size=(6144,16) Running N+1 test using discriminant 53, base 5+sqrt(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) 88128 bit request FFT size=(6144,16) Calling N-1 BLS with factored part 0.44% and helper 0.02% (1.35% proof) (2963^3821-1)/2962 is Fermat and Lucas PRP! (276.0469s+0.0012s) [Elapsed time: 276 seconds]

modified 2020-07-07 22:30:43

created 2005-09-19 15:58:16

id 80802

field	value
prime_id	75630
person_id	9
machine	Linux P4 2.8GHz
what	prp
notes	Command: /home/caldwell/client/pfgw -f -tc -q"(2963^3821-1)/2962" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2963^3821-1)/2962 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 3897723 Running N-1 test using base 7 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) 88120 bit request FFT size=(6144,16) Running N-1 test using base 23 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) 88120 bit request FFT size=(6144,16) Running N-1 test using base 41 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) 88120 bit request FFT size=(6144,16) Running N+1 test using discriminant 53, base 5+sqrt(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) 88128 bit request FFT size=(6144,16) Calling N-1 BLS with factored part 0.44% and helper 0.02% (1.35% proof) (2963^3821-1)/2962 is Fermat and Lucas PRP! (276.0469s+0.0012s) [Elapsed time: 276 seconds]
modified	2020-07-07 22:30:43
created	2005-09-19 15:58:16
id	80802

Query times: 0.0003 seconds to select prime, 0.0004 seconds to seek comments.

(29633821 - 1)/2962

This prime's information:

Archival tags:

User comments about this prime (disclaimer):

Verification data:

(2963³⁸²¹ - 1)/2962