PrimeGrid Generalized Fermat Prime Search

The Generalized Fermat Prime Search is a PrimeGrid sub-project, launched in 2009. It is sometimes abbreviated to "Genefer" or "GFN". It is searching for Generalized Fermat prime numbers.

Purpose

The project is searching for primes of the form b²ⁿ+1 where:

• 15 ≤ n ≤ 22
• b ≤ 2 • 10⁹

"Genefer 17" (where n = 17) was split into two sub-projects:

• "Genefer 17 Low", testing b < 42,597,774, completed.
• "Genefer 17 Mega", testing b ≥ 42,597,774 (the smallest b necessary to find megaprimes)

There is also "Do You Feel Lucky?", a variant of Genefer 22 (where n = 22) designed to find world record primes. It is currently searching b ≥ 846,398, the smallest b necessary such that any prime found would be larger than M51, making it a world record prime number. If a new world record is found outside of PrimeGrid then this project will jump up to the minimum b needed to surpass that number.

Primes found

As of July 2023 over 7,000 primes have been found. Full list from PrimeGrid's website

The largest four finds have been GFN-20s:

• 2022-09-24: 1963736^1048576+1
• 2022-08-09: 1951734^1048576+1
• 2018-10-31: 1059094^1048576+1
• 2017-08-29: 919444^1048576+1

Status

Live status at PrimeGrid

References

PrimeGrid

Miscellaneous

Main

Subprojects

Type Proth : 321 Prime Search 27121 Prime Search Proth Prime Search Proth Prime Search Extended Proth Mega Prime Search Type Sierpiński : Seventeen or Bust Extended Sierpiński Project Prime Sierpiński Problem Sierpiński base 5 Type Riesel : The Riesel Problem Riesel base 5 Type Fermat : Generalized Fermat Prime Search b216+1 b217+1 b218+1 b219+1 b220+1 b221+1 b222+1 Fermat Divisor Search Type Cullen/Woodall : Cullen Prime Search Woodall Prime Search Generalized Cullen Prime Search Type Others : AP27 Search Sophie Germain Search Wieferich and Wall-Sun-Sun Prime Search

Completed

GFN b215+1

Others