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PrimeGrid Generalized Fermat Prime Search
The Generalized Fermat Prime Search is a PrimeGrid subproject, launched in 2009.^{[1]} It is sometimes abbreviated to "Genefer" or "GFN". It is searching for Generalized Fermat prime numbers.
Purpose
The project is currently searching for primes of the form b^{2n}+1 where:
 16 ≤ n ≤ 22
 b ≤ 2 • 10^{9}
PrimeGrid GFN17 (where n = 17) was split into two subprojects:
 PrimeGrid GFN17 Low, testing b < 42,597,774, completed
 PrimeGrid GFN17 Mega, testing b ≥ 42,597,774 (the smallest b necessary to find megaprimes)
There is also "Do You Feel Lucky?", a variant of PrimeGrid GFN22 (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.^{[2]}
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 PrimeGrid GFN20:
 20220924: 1963736^{1048576}+1
 20220809: 1951734^{1048576}+1
 20181031: 1059094^{1048576}+1
 20170829: 919444^{1048576}+1
Status
Completed
 202201: PrimeGrid GFN17 Low, testing b < 42,597,774 (20211114: Forum / 20220123: Post)
 20231115: PrimeGrid GFN15 was terminated at b = 460,000,000. (20230914: Forum / 20231108: Post)
References
Miscellaneous 
Subprojects 

Completed 
Others 