# 23451 · 23739388 + 1

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: 23451 · 23739388 + 1 Proven [none] This prime has 1 user comment below. L591 : Penne, Srsieve, CRUS, LLR 1125673   (log10 is 1125672.3235873) 844 (digit rank is 1) 90 short 8/3/2015 07:04:56 UTC 5/20/2023 20:59:19 UTC 120189 none 46.9962 (normalized score 7.5063)

Jeppe Stig Nielsen writes (15 May 2023):  (report abuse)
 First prime of form 23451*2^{2*m} + 1. This contributes to the proof of the following case of the "Liskovets-Gallot conjectures": Theorem: 66741 is the smallest positive k with k==3 (mod 6) such that no prime of form k*2^n + 1 has an even exponent n. Proof: To see that 66741 has this property, consider the covering set { 5, 7, 13, 17, 241 }. And to see that 66714 is smalles with this property, run through all smaller k with k==3 (mod 6) and find at least one prime of form k*2^{2*m} + 1; the present prime (with k=23451) is the largest one encountered during that calculation.

#### 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.
fieldvalue
prime_id120189
person_id9
machineUsing: Xeon 4c+4c 3.5GHz
whatprime
notesCommand: /home/caldwell/client/pfgw/pfgw64 -t -q"23451*2^3739388+1" 2>&1 PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11] Primality testing 23451*2^3739388+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Calling Brillhart-Lehmer-Selfridge with factored part 100.00% 23451*2^3739388+1 is prime! (6874.1391s+0.0013s) [Elapsed time: 1.91 hours]
modified2020-07-07 22:30:17
created2015-08-03 07:11:01
id165817

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