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:

• Generalized Fermat number - Wikipedia
  
• Generalized Fermat primes - Wikipedia
  
• Generalized Fermat number - Wolfram MathWorld
  
• generalized Fermat number - The Prime Pages
  
• generalized Fermat prime - The Prime Pages
  
• Top 20 Generalized Fermat Primes - The Prime Pages
  

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

• Fermat number - Wikipedia
  
• Fermat Number The Prime Glossary at the Prime Pages
  
• Fermat Number Wolfram Mathworld
  
• Fermat Prime Wolfram Mathworld
  

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

• Pierre de Fermat Wikipedia
  

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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My lucky number is 75898⁵²⁴²⁸⁸+1