# 3^{4043119} + 3^{2021560} + 1

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: | 3^{4043119} + 3^{2021560} + 1 |
---|---|

Verification status (*): | Proven |

Official Comment (*): | Generalized unique |

Unofficial Comments: | This prime has 2 user comments below. |

Proof-code(s): (*): | L5123 : Propper, Batalov, EMsieve, LLR |

Decimal Digits: | 1929059 (log_{10} is 1929058.0102609) |

Rank (*): | 345 (digit rank is 1) |

Entrance Rank (*): | 248 |

Currently on list? (*): | short |

Submitted: | 6/20/2023 18:48:54 UTC |

Last modified: | 3/31/2024 19:30:09 UTC |

Database id: | 136185 |

Status Flags: | none |

Score (*): | 48.648 (normalized score 37.3121) |

#### 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:
yes, rank11

Subcategory: "Generalized Unique"

(archival tag id 228659, tag last modified 2023-12-14 08:37:23)

#### User comments about this prime (disclaimer):

User comments are allowed to convey mathematical information about this number, how it was proven prime.... See our guidelines and restrictions.

#### 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 136185 person_id 9 machine Using: Digital Ocean Droplet what prime notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"Phi(3,3^2021560+1)/3" 2>&1

PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]

Primality testing Phi(3,3^2021560+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge]

trial

Running N-1 test using base 2

Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number

Running N-1 test using base 11

Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number

Running N-1 test using base 13

Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number

Running N+1 test using discriminant 19, base 1+sqrt(19)

Generic modular reduction using generic reduction FMA3 FFT length 672K, Pass1=448, Pass2=1536, clm=4 on A 6408193-bit number

Calling N-1 BLS with factored part 50.00% and helper 0.00% (150.00% proof)

Phi(3,3^2021560+1)/3 is prime! (758315.4225s+0.1827s)

[Elapsed time: 8.78 days]modified 2023-06-29 13:27:37 created 2023-06-20 18:49:01 id 182018

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

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