"((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 (*): | Proven |
| 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 (*): | 2663 (digit rank is 20) |
| Entrance Rank (*): | 1397 |
| Currently on list? (*): | short |
| Submitted: | 5/12/2022 11:45:23 UTC |
| Last modified: | 8/19/2023 11:56:14 UTC |
| Database id: | 133922 |
| Blob database id: | 400 |
| Status Flags: | none |
| Score (*): | 46.633 (normalized score 5.2039) |
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 + 1The 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 1042
Subcategory: "Generalized Fermat"
(archival tag id 227043, tag last modified 2024-04-24 05:37:26)
User comments about this prime (disclaimer):
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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.
field value prime_id 133922 person_id 9 machine Using: Digital Ocean Droplet what prime notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -h133922_helper p_133922.txt 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1000000000...2022291457 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file 133922_helper
trial
Running N-1 test using base 5
Generic modular reduction using generic reduction AVX-512 FFT length 336K, Pass1=896, Pass2=384, clm=1 on A 3321925-bit number
Running N-1 test using base 31
Generic modular reduction using generic reduction AVX-512 FFT length 336K, Pass1=896, Pass2=384, clm=1 on A 3321925-bit number
Running N+1 test using discriminant 43, base 12+sqrt(43)
Generic modular reduction using generic reduction AVX-512 FFT length 336K, Pass1=896, Pass2=384, clm=1 on A 3321925-bit number
Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate
Iteration: 38/3322023 ERROR: ROUND OFF 0.5>0.45
(Test aborted, try again using the -a1 switch)
Running N+1 test using discriminant 43, base 12+sqrt(43)
Generic modular reduction using generic reduction AVX-512 FFT length 360K, Pass1=960, Pass2=384, clm=1 on A 3321925-bit number
Calling N-1 BLS with factored part 100.00% and helper 0.00% (300.00% proof)
1000000000...2022291457 is prime! (259324.1170s+6.1459s)
[Elapsed time: 3.00 days]
Helper File:
41025034033208231128138153244356127...(463 digits)...77954401166576052005843552458560609
2620695441045870361699modified 2023-08-19 11:56:14 created 2023-08-16 11:54:02 id 182188
field value prime_id 133922 person_id 9 machine Using: Dual Intel Xeon Gold 5222 CPUs 3.8GHz what prp notes Command: /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] modified 2022-07-11 18:21:44 created 2022-05-12 11:46:01 id 179644
Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.
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