U(115, - 116, 2089)
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: | U(115, - 116, 2089) |
|---|---|
| Verification status (*): | Proven |
| Official Comment (*): | Generalized Lucas number |
| Proof-code(s): (*): | x25 : Broadhurst, Water, OpenPFGW, Primo |
| Decimal Digits: | 4311 (log10 is 4310.5845536333) |
| Rank (*): | 96771 (digit rank is 1) |
| Entrance Rank (*): | 24306 |
| Currently on list? (*): | no |
| Submitted: | 11/9/2001 17:48:49 UTC |
| Last modified: | 2/4/2026 12:05:31 UTC |
| Database id: | 26968 |
| Status Flags: | none |
| Score (*): | 29.828 (normalized score 0) |
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 Lucas Number (archivable *)
- Prime on list: no, rank 99
Subcategory: "Generalized Lucas Number"
(archival tag id 195498, tag last modified 2025-10-27 15:37:13)
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 26968 person_id 9 machine Using: Digital Ocean Droplet what prime notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -hhelper_1100000000696698412.txt -q"lucasU(115,-116,2089)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing lucasU(115,-116,2089) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000000696698412.txt
trial
Running N-1 test using base 2
Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 14327-bit number
Running N-1 test using base 11
Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 14327-bit number
Running N+1 test using discriminant 31, base 1+sqrt(31)
Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 14327-bit number
Calling N-1 BLS with factored part 36.50% and helper 0.46% (109.98% proof)
lucasU(115,-116,2089) is prime! (1.8678s+0.0003s)
[Elapsed time: 5.00 seconds]
Helper File:
2
3
5
7
19
23
29
37
59
73
181
233
277
349
353
523
1381
1549
2089
4177
4447
6961
13457
16741
22447
32563
33757
59393
69427
129097
131101
241441
512929
515563
1813313
12429127
23985089
32747497
74821051
369835957
1461091249
1860073177
3753210973
15491408791
18696195601
27335628289
911860709113
14533910137843
284582041762289
572111876150281
6186732413950477
11003566298663233
149652812058720121
220044456721397461
8525650228473325817
16977150054110488969
3155408071159372744717
6385558608083061528133
154212426666191820282133
364130753521921737472369
14060152850921008335030640483171
22566718712920855537443875735053
176043718571240933355275482554049
29826749493904638082339111032986780239
1563721650332372051297386247663898037313
155875211976856634561230511869324096119343703989
53894287114905027144409320681720043214186536296849356009040357223929421188711901
16969685298589904598642626670212794...(109 digits)...99704321521125935854576111447309489
34571626573845918564783429269016198...(167 digits)...56856744607152825112929989944807057
43295858560857657567768692021297629...(193 digits)...45290726815701101411322569029339409
67443867787311610918779598689403892...(347 digits)...10390704166206869933422549175357761
509
111767
746124785251modified 2026-02-04 12:05:31 created 2026-02-04 12:05:26 id 187738
field value prime_id 26968 person_id 9 machine Linux PII 200 what prp notes PFGW Version 20020311.x86_Dev (Alpha software, 'caveat utilitor') Primality testing Reading factors from helper file helper Running N-1 test using base 2 1845384465...6995869661 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 11 Running N+1 test using discriminant 31, base 1+sqrt(31) Calling N-1 BLS with factored part 2.51% and helper 0.18% (7.74% proof) U(115,-116,2089) is Fermat and Lucas PRP! (312.100000 seconds) modified 2003-03-25 17:22:46 created 2003-01-07 18:54:48 id 64168
Query times: 0.0003 seconds to select prime, 0.0003 seconds to seek comments.
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