Phi(461, - 6724)
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: | Phi(461, - 6724) |
|---|---|
| Verification status (*): | Proven |
| Official Comment (*): | Generalized unique |
| Proof-code(s): (*): | DB : Dubner, Brent, Cruncher |
| Decimal Digits: | 1761 (log10 is 1760.708744193) |
| Rank (*): | 111755 (digit rank is 1) |
| Entrance Rank (*): | 10010 |
| Currently on list? (*): | no |
| Submitted: | 7/1996 |
| Last modified: | 2/4/2026 10:16:28 UTC |
| Database id: | 38055 |
| Status Flags: | none |
| Score (*): | 27.0396 (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 Unique (archivable *)
- Prime on list: no, rank 1578
Subcategory: "Generalized Unique"
(archival tag id 225622, tag last modified 2026-01-09 22:37:30)
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 38055 person_id 9 machine Using: Digital Ocean Droplet what prime notes Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000001772587607.txt -q"Phi(461,-6724)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(461,-6724) [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000001772587607.txt
Prime_Testing_Warning,
Running N-1 test using base 2
Generic modular reduction using generic reduction FMA3 FFT length 640 on A 5849-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 34.39%
Phi(461,-6724) is prime! (0.1096s+0.0050s)
[Elapsed time: 5.00 seconds]
Helper File:
2
3
5
11
41
47
83
139
191
277
461
881
3313
9431
46691
227471
233861
297641
421361
471041
4160941
4528241
14289211
45212177
69233221
303676561
1916429369
127794114781
214557946741
2119834603241
52473700544521
408767379437801
566376504886781
4589984764943329
1102532018451567881
1589361837607883181091
23588199680819071697881
693607717048646732881656277
988625555339497029880328891
23137009544499169421453572321
57983765914226081958970357804117967
1285978139266094478723147821625123656873807
65711348770408505648715303754969959684929977
134470892758359782550761357642609924994281469143211903555721
41296611573993527644177009393931942589672363367598600021748769792506431
5224350599122589159999392212901243449344377423921287031895654397865392905039521
35843424866816512455774385062762037...(151 digits)...28288070491254737408979741799528371
641493403289modified 2026-02-04 10:16:28 created 2026-02-04 10:16:23 id 187686
field value prime_id 38055 person_id 9 machine Linux PII 200 what prp notes PFGW Version 20020311.x86_Dev (Alpha software, 'caveat utilitor') Running N-1 test using base 2 Primality testing Phi(461,-6724) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 17, base 3+sqrt(17) Calling N-1 BLS with factored part 3.27% and helper 0.02% (9.82% proof) Phi(461,-6724) is Fermat and Lucas PRP! (35.870000 seconds) modified 2003-03-25 17:23:00 created 2003-01-04 17:51:48 id 61457
Query times: 0.0004 seconds to select prime, 0.0005 seconds to seek comments.
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