Phi(3, 32021560 + 1)/3

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, 32021560 + 1)/3
Verification status (*):Proven
Official Comment (*):Generalized unique
Unofficial Comments:This prime has 2 user comments below.
Proof-code(s): (*):L5123 : Propper, Batalov, EMsieve, LLR
Decimal Digits:1929059   (log10 is 1929058.0102609)
Rank (*):311 (digit rank is 1)
Entrance Rank (*):248
Currently on list? (*):short
Submitted:6/20/2023 18:48:54 UTC
Last modified:6/29/2023 13:37:16 UTC
Database id:136185
Status Flags:none
Score (*):48.648 (normalized score 41.7468)

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 11
Subcategory: "Generalized Unique"
(archival tag id 228659, tag last modified 2023-12-14 08:37:23)

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 (21 Jun 2023):  (report abuse)
Eisenstein Mersenne Norm prime #28
This prime may also be written as: 3^4043119+3^2021560+1

Good reference materials for Eisenstein Mersenne Norm primes are at OEIS A066408.

Serge Batalov writes (20 Aug 2023):  (report abuse)
Also, this (and similar numbers with the '+' sign in the middle part) are cuban primes. This one can be represented as 3*y2 + 3*y + 1 where y = 32021559.

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.
fieldvalue
prime_id136185
person_id9
machineUsing: Digital Ocean Droplet
whatprime
notesCommand: /var/www/clientpool/1/pfgw64 -V -f -tc -q"Phi(3,3^2021560+1)/3" 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(3,3^2021560+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial


Running N-1 test using base 2
Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number
Running N-1 test using base 11
Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number
Running N-1 test using base 13
Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number
Running N+1 test using discriminant 19, base 1+sqrt(19)
Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number
Calling N-1 BLS with factored part 50.00% and helper 0.00% (150.00% proof)


Phi(3,3^2021560+1)/3 is prime! (758315.4225s+0.1827s)
[Elapsed time: 8.78 days]
modified2023-06-29 13:27:37
created2023-06-20 18:49:01
id182018

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