5166932097152 - 5166931048576 + 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:5166932097152 - 5166931048576 + 1
Verification status (*):Proven
Official Comment (*):Generalized unique
Unofficial Comments:This prime has 3 user comments below.
Proof-code(s): (*):L4561 : Propper, Batalov, CycloSv, Cyclo, EMsieve, PIES, LLR
Decimal Digits:11981518   (log10 is 11981517.127199)
Rank (*):7 (digit rank is 1)
Entrance Rank (*):7
Currently on list? (*):short
Submitted:10/2/2023 00:29:30 UTC
Last modified:12/14/2023 08:37:20 UTC
Database id:136490
Status Flags:none
Score (*):54.2397 (normalized score 9960.3811)

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 1
Subcategory: "Generalized Unique"
(archival tag id 229036, 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 (2 Oct 2023):  (report abuse)
In Memoriam: Ernst W. Mayer (1963 - 2023)

Serge Batalov writes (2 Oct 2023):  (report abuse)
LLR version >= 4.0.4 Pocklington N-1 proof (using a=31; it takes several passes for other choices of a):
Base factorized as : 3*29*5939
Base prime factor(s) taken : 3, 29, 5939
Found Phi(3,-516693^1048576)...
Starting N-1 prime test of 516693^2097152-516693^1048576+1
Using all-complex AVX FFT length 3M, Pass1=512, Pass2=6K, clm=2, 32 threads, a = 31
516693^2097152-516693^1048576+1 may be prime, trying to compute gcd's
31^((N-1)/5939)-1 is coprime to N!
31^((N-1)/29)-1 is coprime to N!
31^((N-1)/3)-1 is coprime to N!
516693^2097152-516693^1048576+1 is prime! (11981518 decimal digits)  Time : 61973.452 sec.

Serge Batalov writes (10 Nov 2023):  (report abuse)
Using the latest PFGW with GWNUM 30.18b2 that implements Montgomery reduction in a generic modular case:

PFGW Version [GWNUM 30.18]
Primality testing 516693^2097152-516693^1048576+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Generic modular reduction using Montgomery reduction AVX-512 FFT length 2x2160K, Pass1=1152, Pass2=1920, clm=2, 32 threads on A 39801739-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 45.82%
516693^2097152-516693^1048576+1 is prime! (325969.7602s+0.1849s)

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: Digital Ocean Droplet
notesCommand: /var/www/clientpool/1/pfgw64 -V -f -t -q"Phi(3,-516693^1048576)" >command_output 2>&1
PFGW Version [GWNUM 30.11]
Primality testing Phi(3,-516693^1048576) [N-1, Brillhart-Lehmer-Selfridge]

Running N-1 test using base 7
Generic modular reduction using generic reduction FMA3 FFT length 4480K, Pass1=448, Pass2=10K, clm=2 on A 39801739-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 45.82%

Phi(3,-516693^1048576) is prime! (6326575.8607s+1.4961s)
[Elapsed time: 73.22 days]
modified2023-12-14 08:17:51
created2023-10-02 02:54:53

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