# (2^{10691} + 1)/3

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.

Description: | (2^{10691} + 1)/3 |
---|---|

Verification status (*): | PRP |

Official Comment (*): | Generalized Lucas number, Wagstaff, ECPP |

Proof-code(s): (*): | c4 : Broadhurst, Primo |

Decimal Digits: | 3218 (log_{10} is 3217.83456239) |

Rank (*): | 90414 (digit rank is 1) |

Entrance Rank (*): | 37763 |

Currently on list? (*): | short |

Submitted: | 10/13/2004 02:29:59 CDT |

Last modified: | 6/3/2005 13:53:34 CDT |

Database id: | 74682 |

Status Flags: | Verify |

Score (*): | 28.9187 (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, rank105

Subcategory: "Generalized Lucas Number"

(archival tag id 181343, tag last modified 2021-08-03 03:37:35)- Elliptic Curve Primality Proof (archivable *)
- Prime on list:
no, rank587

Subcategory: "ECPP"

(archival tag id 181344, tag last modified 2021-08-03 03:37:33)- Wagstaff (archivable *)
- Prime on list:
yes, rank7

Subcategory: "Wagstaff"

(archival tag id 181345, tag last modified 2021-08-03 03:37:36)

#### 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 74682 person_id 9 machine Linux P4 2.8GHz what prp notes Command: /home/caldwell/client/pfgw -f -tc -q"(2^10691+1)/3" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2^10691+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 839741 Running N-1 test using base 2 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(1280,22) to FFT(1280,21) Reduced from FFT(1280,21) to FFT(1280,20) Reduced from FFT(1280,20) to FFT(1280,19) Reduced from FFT(1280,19) to FFT(1280,18) Reduced from FFT(1280,18) to FFT(1280,17) 21388 bit request FFT size=(1280,17) Running N-1 test using base 5 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(1280,22) to FFT(1280,21) Reduced from FFT(1280,21) to FFT(1280,20) Reduced from FFT(1280,20) to FFT(1280,19) Reduced from FFT(1280,19) to FFT(1280,18) Reduced from FFT(1280,18) to FFT(1280,17) 21388 bit request FFT size=(1280,17) Running N+1 test using discriminant 11, base 1+sqrt(11) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(1280,22) to FFT(1280,21) Reduced from FFT(1280,21) to FFT(1280,20) Reduced from FFT(1280,20) to FFT(1280,19) Reduced from FFT(1280,19) to FFT(1280,18) Reduced from FFT(1280,18) to FFT(1280,17) 21396 bit request FFT size=(1280,17) Running N+1 test using discriminant 11, base 4+sqrt(11) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(1280,22) to FFT(1280,21) Reduced from FFT(1280,21) to FFT(1280,20) Reduced from FFT(1280,20) to FFT(1280,19) Reduced from FFT(1280,19) to FFT(1280,18) Reduced from FFT(1280,18) to FFT(1280,17) 21396 bit request FFT size=(1280,17) Running N+1 test using discriminant 11, base 7+sqrt(11) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(1280,22) to FFT(1280,21) Reduced from FFT(1280,21) to FFT(1280,20) Reduced from FFT(1280,20) to FFT(1280,19) Reduced from FFT(1280,19) to FFT(1280,18) Reduced from FFT(1280,18) to FFT(1280,17) 21396 bit request FFT size=(1280,17) Calling N+1 BLS with factored part 0.42% and helper 0.21% (1.50% proof) (2^10691+1)/3 is Fermat and Lucas PRP! (21.3607s+0.0296s) modified 2020-07-07 17:30:43 created 2005-06-03 16:53:01 id 79724

Query times: 0.0002 seconds to select prime, 0.0004 seconds to seek comments.

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