"26227320567153919098...(166101 other digits)...75512364409752762433"
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: | "26227320567153919098...(166101 other digits)...75512364409752762433" |
|---|---|
| Verification status (*): | PRP |
| Official Comment (*): | [none] |
| Proof-code(s): (*): | p269 : Zhou, OpenPFGW |
| Decimal Digits: | 166141 (log10 is 166140.41875392) |
| Rank (*): | 38794 (digit rank is 2) |
| Entrance Rank (*): | 3807 |
| Currently on list? (*): | no |
| Submitted: | 5/17/2010 16:03:57 UTC |
| Last modified: | 3/11/2023 15:54:10 UTC |
| Removed (*): | 9/30/2010 13:18:23 UTC |
| Database id: | 92686 |
| Blob database id: | 246 |
| Status Flags: | Verify |
| Score (*): | 41.1175 (normalized score 0.0208) |
Description: (from blob table id=246)
By the following definition, this prime is p[1,17] with p[1,0]=3, p[2,0]=5, and p[3,0]=7.
The proof file (2354792 bytes) can be found at http://bitc.bme.emory.edu/~lzhou/prime_certs/pmtrio_3_5_7.18.cert Define:
p[i,j]=ABS[1 + 2 * n[i,j] * p[(i + 1) mod 3,j - 1] * p[(i + 2) mod 3,j - 1]],n is the integer with minimum ABS[n] that makes p[i,j] a prime number.
The primality of p[i,j] can be proven using Brillhart - Lehmer - Selfridge algorithm recursively by using p[(i + 1) mod 3,j - 1] and p[(i + 2) mod 3,j - 1] as helper since n is a small integer, by reducing j to 0.
With this idea, taking
p[1,0]=3, p[2,0]=5, p[3,0]=7 - trival prime numbers.
We got the n[i,j] ( columns : j; rows: i):
i j=1 j=2 j=3 1 1 - 1 - 1 2 - 1 2 - 1 3 - 8 - 10 7 4 - 14 - 3 - 13 5 - 18 24 46 6 24 39 - 32 7 225 - 48 27 8 120 - 76 30 9 - 132 245 - 676 10 316 - 722 65 11 55 - 1197 - 510 12 - 427 - 1716 - 637 13 4651 - 1158 3420 14 - 16337 17640 - 18426 15 - 8915 - 70649 - 31489 16 - 18844 - 92841 124053 17 - 144011 - 8853 - 14042
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 92686 person_id 9 machine Ditto P4 P4 what trial_divided notes Command: /home/ditto/client/pfgw -o -f p_92686.txt 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] 2622732056...1909846293.........8252475512364409 1/1 trial factoring to 58595491 2622732056...9752762433 has no small factor. [Elapsed time: 3213.290 seconds] modified 2020-07-07 22:30:35 created 2010-05-17 17:51:25 id 114756
field value prime_id 92686 person_id 9 machine RedHat P4 P4 what prp notes Command: /home/caldwell/client/pfgw -tc p_92686.txt 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing 2622732056...9752762433 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) Reduced from FFT(65536,18) to FFT(65536,17) 1103822 bit request FFT size=(65536,17) Running N-1 test using base 7 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) Reduced from FFT(65536,18) to FFT(65536,17) 1103822 bit request FFT size=(65536,17) Running N-1 test using base 11 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) Reduced from FFT(65536,18) to FFT(65536,17) 1103822 bit request FFT size=(65536,17) Running N+1 test using discriminant 29, base 1+sqrt(29) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) Reduced from FFT(65536,18) to FFT(65536,17) 1103830 bit request FFT size=(65536,17) Calling N-1 BLS with factored part 0.00% and helper 0.00% (0.01% proof) 2622732056...9752762433 is Fermat and Lucas PRP! (-1633.9138s+0.7600s) [Elapsed time: 6.77 hours] modified 2020-07-07 22:30:35 created 2010-05-18 03:12:00 id 114761
Query times: 0.0002 seconds to select prime, 0.0004 seconds to seek comments.
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