7977227425 · (2368352 - 2257849) + 2110505 + 1

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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:7977227425 · (2368352 - 2257849) + 2110505 + 1
Verification status (*):PRP
Official Comment (*):Consecutive primes arithmetic progression (2,d=6)
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):x52 : Batalov, PolySieve, OpenPFGW, Unknown
Decimal Digits:110895   (log10 is 110894.90281479)
Rank (*):48934 (digit rank is 1)
Entrance Rank (*):48922
Currently on list? (*):yes
Submitted:5/11/2025 20:52:12 UTC
Last modified:5/13/2025 18:37:09 UTC
Database id:140710
Status Flags:Verify
Score (*):39.8728 (normalized score 0.0037)

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.
Consecutive Primes in Arithmetic Progression (archivable class *)
Prime on list: yes, rank 1
Subcategory: "2/2"
(archival tag id 239302, tag last modified 2025-05-13 18:37:10)
Arithmetic Progressions of Primes (archivable class *)
Prime on list: yes, rank 1
Subcategory: "2/2"
(archival tag id 239303, tag last modified 2025-05-13 18:37:10)

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 (12 May 2025):  (report abuse)
Such pairs are called "sexy primes".

Can be written shorter as (7977227425*2^257847+1)*(2^110505-4)+5.
Use helper factors for Konyagin-Pomerance primality proof at >30.0% N-1 proof. (https://web.archive.org/web/20170120062023/http://physics.open.ac.uk/~dbroadhu/cert/kppm.gp)

\r kppm.gp
N=7977227425*(2^368352-2^257849)+2^110505+1;
ls=[2^110505,210299,28452703];
kpm(ls,N)
fraction = 300087/10^6
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5
Round of root: 0
Root OK: above the round
Other roots are complex

Proof completed

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_id140710
person_id9
machineUsing: Digital Ocean Droplet
whatprp
notesCommand: /var/www/clientpool/1/pfgw64 -V -f -t -q"7977227425*(2^368352-2^257849)+2^110505+1" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 7977227425*(2^368352-2^257849)+2^110505+1 [N-1, Brillhart-Lehmer-Selfridge]
trial


Running N-1 test using base 3
Generic modular reduction using generic reduction FMA3 FFT length 36K, Pass1=768, Pass2=48, clm=2 on A 368387-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 30.01%


7977227425*(2^368352-2^257849)+2^110505+1 is PRP! (198.2933s+0.0008s)
[Elapsed time: 3.33 minutes]
modified2025-05-13 17:44:39
created2025-05-13 17:41:19
id186798

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