"((sqrtnint(10999999, 2048) + 2) + 364176)2048 + 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:"((sqrtnint(10999999, 2048) + 2) + 364176)2048 + 1"
Verification status (*):PRP
Official Comment (*):Generalized Fermat
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p417 : Tennant, LLR2, PrivGfnServer, OpenPFGW
Decimal Digits:1000000   (log10 is 999999)
Rank (*):1562 (digit rank is 4)
Entrance Rank (*):1384
Currently on list? (*):short
Submitted:5/12/2022 06:45:23 CDT
Last modified:5/27/2022 15:37:01 CDT
Database id:133922
Blob database id:400
Status Flags:Verify, TrialDiv
Score (*):46.633 (normalized score 7.3882)

Description: (from blob table id=400)

This is GFN - 11 MEGA prime b^2048 + 1, but b is 489 digits long and T5K entries limited to 255 characters. True algebraic representation of the prime:
5825910624014^2048 + 1

The description is in PARI/GP language. sqrtnint(10^999999,2048) + 2 is a minimal value of b required to make number b^2048 + 1 exactly 1,000,000 digits long.

This is a first GFN - 11 MEGA prime. An interesting fact is that 489 - digits base was successfully factorized and number can be proved prime with PFGW. Factorization of the base can be found at http://factordb.com/index.php?id=1100000002986607281

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 450
Subcategory: "Generalized Fermat"
(archival tag id 227043, tag last modified 2022-09-24 15:50:10)

User comments about this prime (disclaimer):

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Jeppe Stig Nielsen writes (14 May 2022):  (report abuse)

Smallest megaprime of form b^2048 + 1.

b can be given as sqrtnint(10^999999, 2048) + 364178, or floor(10^(999999/2048)) + 364178.

The full expansion of b is:

Surprisingly, it is relatively easy to factor b completely:
2 * 3 * 11 * 269 * 2620695441045870361699 * P463
This factorization is used for the N-1 primality proof of b^2048 + 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: Dual Intel Xeon Gold 5222 CPUs 3.8GHz
notesCommand: /home/caldwell/clientpool/1/pfgw64 -tc p_133922.txt 2>&1 PFGW Version [GWNUM 29.8] Primality testing 1000000000...2022291457 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Running N+1 test using discriminant 31, base 1+sqrt(31) Calling N-1 BLS with factored part 0.87% and helper 0.00% (2.61% proof) 1000000000...2022291457 is Fermat and Lucas PRP! (56612.7899s+1.9921s) [Elapsed time: 15.73 hours]
modified2022-07-11 13:21:44
created2022-05-12 06:46:01

Query times: 0.0003 seconds to select prime, 0.0004 seconds to seek comments.
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