215317227 + 27658614 + 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:215317227 + 27658614 + 1
Verification status (*):Proven
Official Comment (*):Gaussian Mersenne norm 41?, generalized unique
Unofficial Comments:This prime has 2 user comments below.
Proof-code(s): (*):L5123 : Propper, Batalov, EMsieve, LLR
Decimal Digits:4610945   (log10 is 4610944.7773942)
Rank (*):51 (digit rank is 1)
Entrance Rank (*):20
Currently on list? (*):short
Submitted:7/31/2020 21:56:46 UTC
Last modified:3/11/2023 15:54:10 UTC
Database id:131043
Status Flags:none
Score (*):51.3175 (normalized score 549.3571)

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.
Gaussian Mersenne norm (archivable *)
Prime on list: yes, rank 1
Subcategory: "Gaussian Mersenne norm"
(archival tag id 224076, tag last modified 2023-03-11 15:53:59)
Generalized Unique (archivable *)
Prime on list: yes, rank 4
Subcategory: "Generalized Unique"
(archival tag id 224078, 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 (31 Jul 2020):  (report abuse)
This is also a Generalized Unique number (all Gaussian Mersenne norms are, as well).
To recognize the Generalized Unique property, we can write this number as:
Phi(4, 27658614 + 1)/2

Serge Batalov writes (31 Jul 2020):  (report abuse)
A note on double-checking: one may want to use a 4-/8-threaded LLR with the following input (this will take care of a much faster FFT modulus):
ABC 4^$a+1

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"2^15317227+2^7658614+1" 2>&1 PFGW Version [GWNUM 29.8] Primality testing 2^15317227+2^7658614+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Calling Brillhart-Lehmer-Selfridge with factored part 50.00% 2^15317227+2^7658614+1 is prime! (750648.5746s+0.0068s) [Elapsed time: 8.69 days]
modified2021-04-20 22:39:26
created2020-07-31 22:01:01

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