27 · 2672007 + 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:27 · 2672007 + 1
Verification status (*):Proven
Official Comment (*):Divides Fermat F(672005)
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):g279 : Cooper, NewPGen, PRP, Proth.exe
Decimal Digits:202296   (log10 is 202295.69565993)
Rank (*):31493 (digit rank is 1)
Entrance Rank (*):192
Currently on list? (*):short
Submitted:8/30/2005 02:54:48 UTC
Last modified:3/11/2023 15:54:10 UTC
Database id:75472
Status Flags:none
Score (*):41.7234 (normalized score 0.0365)

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.
Fermat Divisors (archivable *)
Prime on list: yes, rank 20, weight 45.019295902982
Subcategory: "Divides Fermat"
(archival tag id 187210, tag last modified 2023-07-28 18:37:27)

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.

Curtis Cooper writes (11 Sep 2014):  (report abuse)
27*2^672007+1 is a Factor of F672005
27*2^672007+1 is a Factor of xGF(672006,7,3)
27*2^672007+1 is a Factor of xGF(672006,7,6)
27*2^672007+1 is a Factor of xGF(672005,11,3)
27*2^672007+1 is a Factor of xGF(672004,11,6)
27*2^672007+1 is a Factor of xGF(672006,11,7)
27*2^672007+1 is a Factor of xGF(672006,12,7)
27*2^672007+1 is a Factor of xGF(672005,12,11)

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.
machineLinux P4 2.8GHz
notesCommand: /home/caldwell/client/pfgw -f -t -q"27*2^672007+1" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing 27*2^672007+1 [N-1, Brillhart-Lehmer-Selfridge] trial factoring to 72273173 Running N-1 test using base 5 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(98304,20) to FFT(98304,19) Reduced from FFT(98304,19) to FFT(98304,18) Reduced from FFT(98304,18) to FFT(98304,17) Reduced from FFT(98304,17) to FFT(98304,16) 1344032 bit request FFT size=(98304,16) Calling Brillhart-Lehmer-Selfridge with factored part 100.00% 27*2^672007+1 is prime! (9915.2392s+0.0506s) [Elapsed time: 9916 seconds]
modified2020-07-07 22:30:43
created2005-08-30 03:23:00

Query times: 0.0002 seconds to select prime, 0.0014 seconds to seek comments.
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