108416458051\
3253166678679240531131941884663704319991773199919016384 + 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:108416458051\
3253166678679240531131941884663704319991773199919016384 + 1
Verification status (*):Proven
Official Comment (*):Generalized Fermat
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):L5749 : Gahan, LLR2, LLR
Decimal Digits:1000000   (log10 is 999999)
Rank (*):2885 (digit rank is 9)
Entrance Rank (*):2067
Currently on list? (*):yes
Submitted:5/2/2023 15:33:21 UTC
Last modified:5/20/2023 20:59:19 UTC
Database id:136011
Status Flags:none
Score (*):46.633 (normalized score 4.5047)

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 Fermat (archivable *)
Prime on list: no, rank 1170
Subcategory: "Generalized Fermat"
(archival tag id 228534, tag last modified 2024-11-21 13:37:10)

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.

Jeppe Stig Nielsen writes (2 May 2023):  (report abuse)

Is of form b^16384 + 1 where b has 62 digits.

b can be written a bit more compactly as sqrtnint(10^999999, 16384) + 855112 or floor(10^(999999/16384)) + 855112.

The full factorization of the base b is 2 * 3^2 * 5 * 541 * 827 * 273706197559437111817 * 983705952706871468858923621002289. So to rigorously prove this number prime, make a helper file containing at least 

273706197559437111817
983705952706871468858923621002289

and call PFGW with the -h switch.

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_id136011
person_id9
machineUsing: Digital Ocean Droplet
whatprime
notesPFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1084164580...7731999190^16384+1 [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_file_id_136011
Running N-1 test using base 7
Generic modular reduction using generic reduction AVX-512 FFT length 336K, Pass1=896, Pass2=384, clm=1 on A 3321925-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 54.06%


1084164580...7731999190^16384+1 is prime! (27116.8015s+0.0215s)
[Elapsed time: 7.53 hours]
Helper file contains: "273706197559437111817,983705952706871468858923621002289"
modified2023-05-05 06:06:33
created2023-05-04 22:34:36
id181835

fieldvalue
prime_id136011
person_id9
machineUsing: Digital Ocean Droplet
whatprp
notesCommand: /var/www/clientpool/1/pfgw64 -V -t -q"10841645805132531666786792405311319418846637043199917731999190^16384+1" 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1084164580...7731999190^16384+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Generic modular reduction using generic reduction AVX-512 FFT length 336K, Pass1=896, Pass2=384, clm=1 on A 3321925-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 12.46%


1084164580...7731999190^16384+1 is PRP! (19830.1946s+0.0182s)
[Elapsed time: 5.51 hours]
modified2023-05-02 21:04:32
created2023-05-02 15:34:01
id181825

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