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Message boards : Generalized Fermat Prime Search : Welcome to the Generalized Fermat Prime Search

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Profile Michael GoetzProject donor
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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,822,730)Cullen LLR Ruby: Earned 2,000,000 credits (2,005,249)ESP LLR Turquoise: Earned 5,000,000 credits (5,009,577)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,145,754)PPS LLR Jade: Earned 10,000,000 credits (12,843,762)PSP LLR Turquoise: Earned 5,000,000 credits (5,197,957)SoB LLR Sapphire: Earned 20,000,000 credits (34,291,181)SR5 LLR Jade: Earned 10,000,000 credits (10,004,590)SGS LLR Ruby: Earned 2,000,000 credits (2,276,011)TRP LLR Ruby: Earned 2,000,000 credits (2,737,347)Woodall LLR Ruby: Earned 2,000,000 credits (2,195,123)321 Sieve Turquoise: Earned 5,000,000 credits (9,403,130)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (20,114,159)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,114,260)GFN Emerald: Earned 50,000,000 credits (72,383,585)PSA Jade: Earned 10,000,000 credits (12,404,447)
Message 75997 - Posted: 29 Apr 2014 | 7:14:18 UTC
Last modified: 28 May 2018 | 18:35:32 UTC

Welcome to the Generalized Fermat Prime Search

This search is for primes of the form b^2^n+1. The numbers F(b,n) = b^2^n+1 (with n and b integers, b greater than one) are called generalized Fermat numbers. In the special case where b=2, they are called Fermat numbers, named after Pierre de Fermat who first studied them.

The original Generalized Fermat Prime Search by Yves Gallot was very active from 2001-2004. It was a premier project ranking second only to GIMPS in organization and size of primes found. In 2009, PrimeGrid, through its PRPNet, revitalized the search thanks in large part to David Underbakke, Mark Rodenkirch, and Shoichiro Yamada, each of whom provided the necessary software updates to get the project moving again. In 2012 Michael Goetz's native BOINC port of a modified GeneferCUDA, allowed for moving this project to BOINC and greatly accelerating the search. In 2015, Yves Gallot (with some help from Iain Bethune) wrote new versions of Genefer that let us search higher "b" ranges than had previously been possible, allowed the use of all GPUs instead of just Nvidia GPUs, and significantly improved the speed of the applications. As a result, all n's from 15 (32768) through 22 (4194304) are now searched via BOINC.


GFN prime searches available at PrimeGrid

PrimeGrid offers the following GFN prime searches:

Via BOINC
N=32768 (n=15)
N=65536 (n=16)
N=131072 (n=17)
N=131072 (n=17), starting with b=42,597,774. (Special GFN-17 mega prime search)
N=262144 (n=18)
N=524288 (n=19)
N=1048576 (n=20)
N=2097152 (n=21)
N=4194304 (n=22)


The search for N=2097152 (n=21) has the potential of discovering a top 10 prime, while the search for N=4194304 (n=22) additionally has the potential of discovering the world's third largest known prime number. Due to the size of the work units at N=22, this search is a GPU-only project.

Software/Hardware requirements

Generally speaking, any discreet GPU can be used, and any CPU can be used. Certain transforms have special GPU requirements, but unless you have a particularly old GPU this shouldn't be a concern. For details, read the specifics of the different transforms at the beginning of this forum post.

Best of Luck to everyone!!!


GFN sieving

A search such as this obviously needs a big sieve effort. If you would like to help out with the manual sieving effort, please see manual sieving. It's available for GPUs ONLY, under Windows and Linux.

For more information about generalized Fermat numbers and primes, please visit these links:

For more information about Fermat numbers and primes, please visit these links:

For more information about Pierre de Fermat, please visit these links:

A special thanks to the following people:


  • Yves Gallot for the original Genefer program, the original GFN search, and the new updates to Genefer. None of this would be possible without Yves.
  • David Underbakke for AthGfn64 (sieve), Genefx64 and updates to Genefer, and Genefer80 used in the PRPNet searches.
  • Mark Rodenkirch for PRPNet and his collaboration with David in updating Genefx64, Genefer, and Genefer80 to prepare the programs for a distributed effort.
  • Shoichiro Yamada for GeneferCUDA.
  • Michael Goetz for the native BOINC build of GeneferCUDA - Windows.
  • Iain Bethune for the native BOINC build of GenefX64.
  • Ronald Schneider for the native BOINC build of GeneferCUDA - Linux.

  • and to Everyone who attempts this search. :)


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Message boards : Generalized Fermat Prime Search : Welcome to the Generalized Fermat Prime Search

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