U(1731, 1, 21000) - U(1731, 1, 20999)

Previous Next Random

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(1731, 1, 21000) - U(1731, 1, 20999)
Verification status (*):	PRP
Official Comment (*):	Lehmer number
Unofficial Comments:	This prime has 1 user comment below.
Proof-code(s): (*):	p452 : Propper, Batalov, CM, OpenPFGW
Decimal Digits:	68001 (log₁₀ is 68000.996833885)
Rank (*):	58340 (digit rank is 1)
Entrance Rank (*):	58332
Currently on list? (*):	yes
Submitted:	10/3/2025 22:43:29 UTC
Last modified:	10/4/2025 09:37:09 UTC
Database id:	141122
Status Flags:	Verify
Score (*):	38.3656 (normalized score 0.0006)

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 number (archivable *)
Prime on list: yes, rank 4
Subcategory: "Lehmer number"
(archival tag id 239542, tag last modified 2025-10-05 02:37:14)

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.

Serge Batalov writes (3 Oct 2025): (report abuse)

This Lehmer prime number is proven using BLS with N-1 factored to 42.9%. p29210=gcd(primU(1731,1,20999),lucasU(1731,1,10500)+lucasU(1731,1,10499)), a Lehmer primitive part prime, is proven with CM. Ryan Propper provided massive PRP screening.

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 141122

person_id 9

machine Using: Digital Ocean Droplet

what prp

notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"lucasU(1731,1,21000)-lucasU(1731,1,20999)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing lucasU(1731,1,21000)-lucasU(1731,1,20999) [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial

Running N-1 test using base 17
Generic modular reduction using generic reduction FMA3 FFT length 24K, Pass1=384, Pass2=64, clm=2 on A 225895-bit number
Running N+1 test using discriminant 31, base 24+sqrt(31)
Generic modular reduction using generic reduction FMA3 FFT length 24K, Pass1=384, Pass2=64, clm=2 on A 225895-bit number
Calling N-1 BLS with factored part 0.41% and helper 0.16% (1.38% proof)

lucasU(1731,1,21000)-lucasU(1731,1,20999) is Fermat and Lucas PRP! (308.1998s+0.0043s)
[Elapsed time: 5.18 minutes]

modified 2025-10-04 09:31:45

created 2025-10-04 09:26:34

id 187212

field	value
prime_id	141122
person_id	9
machine	Using: Digital Ocean Droplet
what	prp
notes	Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"lucasU(1731,1,21000)-lucasU(1731,1,20999)" >command_output 2>&1 PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11] Primality testing lucasU(1731,1,21000)-lucasU(1731,1,20999) [N-1/N+1, Brillhart-Lehmer-Selfridge] trial Running N-1 test using base 17 Generic modular reduction using generic reduction FMA3 FFT length 24K, Pass1=384, Pass2=64, clm=2 on A 225895-bit number Running N+1 test using discriminant 31, base 24+sqrt(31) Generic modular reduction using generic reduction FMA3 FFT length 24K, Pass1=384, Pass2=64, clm=2 on A 225895-bit number Calling N-1 BLS with factored part 0.41% and helper 0.16% (1.38% proof) lucasU(1731,1,21000)-lucasU(1731,1,20999) is Fermat and Lucas PRP! (308.1998s+0.0043s) [Elapsed time: 5.18 minutes]
modified	2025-10-04 09:31:45
created	2025-10-04 09:26:34
id	187212

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