(2136883 - 1)/536581361
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: | (2136883 - 1)/536581361 |
|---|---|
| Verification status (*): | PRP |
| Official Comment (*): | Mersenne cofactor, ECPP |
| Proof-code(s): (*): | E5 : Underwood, CM |
| Decimal Digits: | 41198 (log10 is 41197.15926089) |
| Rank (*): | 66744 (digit rank is 1) |
| Entrance Rank (*): | 66593 |
| Currently on list? (*): | yes |
| Submitted: | 8/16/2025 20:03:21 UTC |
| Last modified: | 8/21/2025 04:37:11 UTC |
| Database id: | 141009 |
| Status Flags: | Verify |
| Score (*): | 36.8193 (normalized score 0.0001) |
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.
- Elliptic Curve Primality Proof (archivable *)
- Prime on list: yes, rank 18
Subcategory: "ECPP"
(archival tag id 239473, tag last modified 2025-09-08 02:37:14)- Mersenne cofactor (archivable *)
- Prime on list: yes, rank 4
Subcategory: "Mersenne cofactor"
(archival tag id 239474, tag last modified 2025-09-08 02:37:15)
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 141009 person_id 9 machine Using: Digital Ocean Droplet what prp notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"(2^136883-1)/536581361" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing (2^136883-1)/536581361 [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial
Running N-1 test using base 5
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 136855-bit number
Running N+1 test using discriminant 11, base 2+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 136855-bit number
Calling N+1 BLS with factored part 0.03% and helper 0.02% (0.10% proof)
(2^136883-1)/536581361 is Fermat and Lucas PRP! (146.1302s+0.0007s)
[Elapsed time: 2.50 minutes]modified 2025-08-21 04:09:19 created 2025-08-21 04:06:49 id 187098
Query times: 0.0002 seconds to select prime, 0.0002 seconds to seek comments.
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