2167681045402 · 4663# + 3399421637
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: | 2167681045402 · 4663# + 3399421637 |
|---|---|
| Verification status (*): | PRP |
| Official Comment (*): | Consecutive primes arithmetic progression (5,d=30) |
| Proof-code(s): (*): | E14 : Batalov, EMsieve, CM |
| Decimal Digits: | 2006 (log10 is 2005.4174819235) |
| Rank (*): | 110580 (digit rank is 135) |
| Entrance Rank (*): | 110568 |
| Currently on list? (*): | yes |
| Submitted: | 6/10/2026 17:04:32 UTC |
| Last modified: | 6/12/2026 21:37:11 UTC |
| Database id: | 142309 |
| Status Flags: | Verify |
| Score (*): | 27.4456 (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.
- Consecutive Primes in Arithmetic Progression (archivable class *)
- Prime on list: yes, rank 1
Subcategory: "5/5"
(archival tag id 240328, tag last modified 2026-06-12 21:37:13)- Arithmetic Progressions of Primes (archivable class *)
- Prime on list: no, rank 151, weight 40.430106924762
Subcategory: "5/5"
(archival tag id 240329, tag last modified 2026-06-12 21:37:15)
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 142309 person_id 9 machine Using: Digital Ocean Droplet what prp notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"2167681045402*4663#+3399421637" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 2167681045402*4663#+3399421637 [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial
Running N-1 test using base 3
Generic modular reduction using generic reduction FMA3 FFT length 640 on A 6662-bit number
Running N-1 test using base 5
Generic modular reduction using generic reduction FMA3 FFT length 640 on A 6662-bit number
Running N+1 test using discriminant 11, base 1+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 640 on A 6662-bit number
Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate
Iteration: 5219/6679 ERROR: ROUND OFF 0.5>0.45
(Test aborted, try again using the -a1 switch)
Running N+1 test using discriminant 11, base 1+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 768 on A 6662-bit number
Calling N-1 BLS with factored part 0.72% and helper 0.12% (2.31% proof)
2167681045402*4663#+3399421637 is Fermat and Lucas PRP! (0.3527s+0.0004s)
[Elapsed time: 5.00 seconds]modified 2026-06-12 20:49:48 created 2026-06-12 20:49:43 id 188495
Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.
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