31723 · 2^1398273 - 507567

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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:	31723 · 2^1398273 - 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 (*):	12965 (digit rank is 2)
Entrance Rank (*):	1070
Currently on list? (*):	no
Submitted:	9/6/2013 19:11:39 UTC
Last modified:	5/20/2023 20:59:19 UTC
Removed (*):	3/17/2019 12:32:19 UTC
Database id:	115395
Status Flags:	none
Score (*):	43.9762 (normalized score 0.3647)

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*M_p+1) could have been a divisor for the double-Mersenne number 2^M_p-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.

field value

prime_id 115395

person_id 9

machine WinXP Dual Core 2.6GHz 32-bit

what prime

notes Command: pfgw32.exe -tc -hhelp.txt -q"31723*2^1398273-507567" 2>&1 PFGW Version 3.7.3.32BIT.20130210.Win_Dev [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

modified 2020-07-07 22:30:18

created 2013-09-10 15:15:54

id 160939

field	value
prime_id	115395
person_id	9
machine	WinXP Dual Core 2.6GHz 32-bit
what	prime
notes	Command: pfgw32.exe -tc -hhelp.txt -q"317232^1398273-507567" 2>&1 PFGW Version 3.7.3.32BIT.20130210.Win_Dev [GWNUM 27.8] Primality testing 317232^1398273-507567 [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
modified	2020-07-07 22:30:18
created	2013-09-10 15:15:54
id	160939

Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.

31723 · 21398273 - 507567

This prime's information:

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Verification data:

31723 · 2^1398273 - 507567