(3040612288 + 698989 · (304068192 + 1))2 + 2
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GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly. This list is the most important databases at The Prime Pages: 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:

field (help)value
Description:(3040612288 + 698989 · (304068192 + 1))2 + 2
Verification status (*):Proven
Official Comment:
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p192 : Gallot, Broadhurst, Proth.exe, OpenPFGW
Decimal Digits:110174   (log10 is 110173.207539726)
Rank (*):39339 (digit rank is 2)
Entrance Rank (*):1166
Currently on list? (*):no
Submitted:4/17/2006 04:16:09 CDT
Last modified:7/7/2006 04:39:47 CDT
Removed (*):3/5/2009 23:46:38 CDT
Database id:77601
Status Flags:none
Score (*):39.8527 (normalized score 0.0144)

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): 
Near-square prime with a BLS proof helped by the smallest GFN-13 that is prime, namely 30406^(2^13)+1, found and proven by Yves Gallot, in February 1999.

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 Pent_M 1.7GHz Laptop
notesCommand: pfgw.exe -n -tc -hhelp.txt -q"(30406^12288+698989*(30406^8192+1))^2+2" 2>&1
PFGW Version 20030811.Win_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]
Reading factors from helper file help.txt
Running N-1 test using base 2
Primality testing (30406^12288+698989*(30406^8192+1))^2+2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Running N+1 test using discriminant 5, base 2+sqrt(5)
Running N+1 test using discriminant 5, base 4+sqrt(5)
N+1: (30406^12288+698989*(30406^8192+1))^2+2 302500/365988Calling N-1 BLS with factored part 33.34% and helper 0.01% (100.02% proof)
(30406^12288+698989*(30406^8192+1))^2+2 is prime! (83631.4790s+0.1958s)
[Elapsed time: 83628 seconds]
Helper File:
modified2006-07-14 06:27:06
created2006-04-18 07:28:36

machineGenToo P3 400MHz
notesCommand: /home/caldwell/client/pfgw -o -f -q"(30406^12288+698989*(30406^8192+1))^2+2" 2>&1
PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]
trial factoring to 37804219
(30406^12288+698989*(30406^8192+1))^2+2 has no small factor.
[Elapsed time: 315.396 seconds]
modified2006-07-14 06:27:04
created2006-07-07 04:34:32

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