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# PrimeGrid Generalized Fermat Prime Search

The Generalized Fermat Prime Search is a PrimeGrid sub-project, launched in 2009.[1] 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 in January 2022.[2]
• "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.[3]

"Genefer 15", the n = 15 search, was terminated in November 2023 at b = 4.6 • 108.[4]

## 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: 19637361048576+1
• 2022-08-09: 19517341048576+1
• 2018-10-31: 10590941048576+1
• 2017-08-29: 9194441048576+1

## References

PrimeGrid
 Miscellaneous
 Subprojects Type Riesel : The Riesel Problem Riesel base 5 Type Fermat : Generalized Fermat Prime Search Fermat Divisor Search Type Cullen/Woodall : Cullen Prime Search Woodall Prime Search Generalized Cullen Prime Search
 Completed ...
 Others