31723 · 21398273 - 507567

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:31723 · 21398273 - 507567
Verification status (*):Proven
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p363 : Batalov, OpenPFGW
Decimal Digits:420927   (log10 is 420926.61650131)
Rank (*):9524 (digit rank is 2)
Entrance Rank (*):1070
Currently on list? (*):no
Submitted:9/6/2013 14:11:39 CDT
Last modified:10/4/2013 13:02:05 CDT
Removed (*):3/17/2019 07:32:19 CDT
Database id:115395
Status Flags:TrialDiv
Score (*):43.9762 (normalized score 0.5184)

User comments about this prime (disclaimer):

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Serge Batalov writes (11 Sep 2014):  (report abuse)
Also can be written as 507568*(2^1398269-1)+1
Use Mersenne#35 (2^1398269-1) as a helper prime for the N-1 proof

With a very low probability, a similar prime (i.e. 2*k*Mp+1) could have been a divisor for the double-Mersenne number 2Mp-1, but this one isn't.

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.
machineWinXP Dual Core 2.6GHz 32-bit
notesCommand: pfgw32.exe -tc -hhelp.txt -q"31723*2^1398273-507567" 2>&1 PFGW Version [GWNUM 27.8] Primality testing 31723*2^1398273-507567 [N-1/N+1, Brillhart-Lehmer-Selfridge] Reading factors from helper file help.txt Sieve re-allocated with a limit of 400000000000. Running N-1 test using base 3 Running N-1 test using base 13 Running N+1 test using discriminant 19, base 2+sqrt(19) Calling N-1 BLS with factored part 100.00% and helper 0.00% (300.00% proof) 31723*2^1398273-507567 is prime! (133555.7758s+0.0035s) [Elapsed time: 133553 seconds] Helper File: 2^1398269-1
modified2020-07-07 17:30:18
created2013-09-10 10:15:54

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