Phi(3, - 143332393216)

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:Phi(3, - 143332393216)
Verification status (*):Proven
Official Comment (*):Generalized unique
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):L4506 : Propper, Batalov, CycloSv, EMsieve, PIES, Prime95, LLR
Decimal Digits:4055114   (log10 is 4055113.264707)
Rank (*):53 (digit rank is 1)
Entrance Rank (*):12
Currently on list? (*):short
Submitted:1/30/2017 05:56:38 CDT
Last modified:2/6/2017 06:23:19 CDT
Database id:122812
Status Flags:TrialDiv
Score (*):50.9242 (normalized score 506.674)

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.
Generalized Unique (archivable *)
Prime on list: yes, rank 3
Subcategory: "Generalized Unique"
(archival tag id 224079, tag last modified 2020-08-04 13:50:04)

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.

Serge Batalov writes (30 Jan 2017):  (report abuse)
Using generic reduction FMA3 FFT length 1440K, Pass1=320, Pass2=4608, 18 threads, a = 11
11^((N-1)/5119)-1 is coprime to N!
11^((N-1)/7)-1 is coprime to N!
11^((N-1)/2)-1 is coprime to N!
143332^786432-143332^393216+1 is prime! (4055114 decimal digits)  Time : 105281.520 sec.

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.
machineUsing: Xeon 4c+4c 3.5GHz
notesCommand: /home/caldwell/client/pfgw/pfgw64 -t -q"Phi(3,-143332^393216)" 2>&1 PFGW Version [GWNUM 27.11] Primality testing Phi(3,-143332^393216) [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 11 Calling Brillhart-Lehmer-Selfridge with factored part 35.97% Phi(3,-143332^393216) is prime! (606135.7958s+1.1064s) [Elapsed time: 7.02 days]
modified2020-07-07 17:30:16
created2017-01-30 06:01:01

Query times: 0.0002 seconds to select prime, 0.0004 seconds to seek comments.
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