"((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 (*):1498 (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 7.5035)

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:
19088056804361919674585871086919934026273973916529754128886026191923
13005206370260816731962602517640847305729238216802572429540910611851
12017397561405538291724020883036948742802421424996617325672792384599
99246507207836394789392438941597163797180877415520372798108941100064
91708543004139014341404449003423872238143355925810943102317611824872
16832411502606252990491877497417433401295278390481355152602752881480
79345041549498152750876354299491043111004326040615308071006479110027
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 433
Subcategory: "Generalized Fermat"
(archival tag id 227043, tag last modified 2022-08-04 00:37:09)

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:

190880568043619196745858710869199340262739739165297541288860261919231
300520637026081673196260251764084730572923821680257242954091061185112
017397561405538291724020883036948742802421424996617325672792384599992
465072078363947893924389415971637971808774155203727981089411000649170
854300413901434140444900342387223814335592581094310231761182487216832
411502606252990491877497417433401295278390481355152602752881480793450
415494981527508763542994910431110043260406153080710064791100275825910
624014
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.
fieldvalue
prime_id133922
person_id9
machineUsing: Dual Intel Xeon Gold 5222 CPUs 3.8GHz
whatprp
notesCommand: /home/caldwell/clientpool/1/pfgw64 -tc p_133922.txt 2>&1 PFGW Version 4.0.1.64BIT.20191203.x86_Dev [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
id179644

Query times: 0.0005 seconds to select prime, 0.0005 seconds to seek comments.
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