(U(9275, 1, 3961) + U(9275, 1, 3960))/(U(9275, 1, 45) + U(9275, 1, 44))

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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:	(U(9275, 1, 3961) + U(9275, 1, 3960))/(U(9275, 1, 45) + U(9275, 1, 44))
Verification status (*):	PRP
Official Comment (*):	Lehmer primitive part
Unofficial Comments:	This prime has 1 user comment below.
Proof-code(s): (*):	x38 : Broadhurst, OpenPFGW, Primo
Decimal Digits:	15537 (log₁₀ is 15536.001284243)
Rank (*):	77357 (digit rank is 2)
Entrance Rank (*):	35575
Currently on list? (*):	yes
Submitted:	5/10/2009 09:19:03 UTC
Last modified:	3/11/2023 15:54:10 UTC
Database id:	88162
Status Flags:	Verify
Score (*):	33.8047 (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.

Lehmer primitive part (archivable *)
Prime on list: yes, rank 20
Subcategory: "Lehmer primitive part"
(archival tag id 210442, tag last modified 2023-03-11 15:53:59)

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.

David Broadhurst writes (11 Sep 2014): (report abuse)

BLS tests were performed with factorization fractions of 31.10% in N-1 and 8.65% in N+1. The proof depends upon the primality of helpers with 638, 1034 and 3490 digits, proven by Primo. Because the index 3961+3960=89^2 is the square of a prime, this Lehmer primitive part is also the unique prime in the Lehmer sequence U(P,1,45)+U(P,1,44) with P=V(9275,1,89). At time of proof, it was the largest known unique Lehmer prime and also the largest known prime that is a Lehmer primitive part.

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 88162

person_id 9

machine RedHat P4 P4

what prp

notes Command: /home/caldwell/client/pfgw -tc -q"(lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44))" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 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) 103228 bit request FFT size=(6144,17) 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) 103228 bit request FFT size=(6144,17) Running N+1 test using discriminant 23, base 1+sqrt(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) 103236 bit request FFT size=(6144,17) Calling N-1 BLS with factored part 0.57% and helper 0.29% (2.01% proof) (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) is Fermat and Lucas PRP! (150.4400s+0.0100s) [Elapsed time: 2.77 minutes]

modified 2020-07-07 22:30:38

created 2009-05-10 09:23:01

id 105686

field	value
prime_id	88162
person_id	9
machine	RedHat P4 P4
what	prp
notes	Command: /home/caldwell/client/pfgw -tc -q"(lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44))" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 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) 103228 bit request FFT size=(6144,17) 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) 103228 bit request FFT size=(6144,17) Running N+1 test using discriminant 23, base 1+sqrt(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) 103236 bit request FFT size=(6144,17) Calling N-1 BLS with factored part 0.57% and helper 0.29% (2.01% proof) (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) is Fermat and Lucas PRP! (150.4400s+0.0100s) [Elapsed time: 2.77 minutes]
modified	2020-07-07 22:30:38
created	2009-05-10 09:23:01
id	105686

field value

prime_id 88162

person_id 9

machine Ditto P4 P4

what trial_divided

notes Command: /home/ditto/client/pfgw -o -f -q"(lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44))" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] (lucasU(9275,1,3961)+luca.........,1,45)+lucasU(92 1/1 trial factoring to 4623227 (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) has no small factor. [Elapsed time: 6.252 seconds]

modified 2020-07-07 22:30:38

created 2009-05-10 09:35:01

id 105687

field	value
prime_id	88162
person_id	9
machine	Ditto P4 P4
what	trial_divided
notes	Command: /home/ditto/client/pfgw -o -f -q"(lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44))" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] (lucasU(9275,1,3961)+luca.........,1,45)+lucasU(92 1/1 trial factoring to 4623227 (lucasU(9275,1,3961)+lucasU(9275,1,3960))/(lucasU(9275,1,45)+lucasU(9275,1,44)) has no small factor. [Elapsed time: 6.252 seconds]
modified	2020-07-07 22:30:38
created	2009-05-10 09:35:01
id	105687

Query times: 0.0002 seconds to select prime, 0.0004 seconds to seek comments.