Generalized Fermat Numbers


There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form a(2n)+1 with a>2, while Riesel (1994) further generalizes, defining it to be a number of the form a(2n)+b(2n). Both definitions generalize the usual Fermat numbers F_n = 2(2n)+1.

Generalized Fermat numbers can be prime only for even a. More specifically, an odd prime p is a generalized Fermat prime iff there exists an integer i with i2 ≡ -1 (mod p) and i2 < p (Broadhurst 2006).

(Weisstein, Eric W. "Generalized Fermat Number." From MathWorld--A Wolfram Web Resource.


Generalized Fermat Primes

From the 50's to the 70's, the size of the largest known prime has grown continuously with the speed of computers but the algorithms used to find them were the same as those used at the end of the 19th century. But in the 80's, the method used to compute the basic operation of the algorithm, the multiplication, changed. By remarking that a multiplication can be evaluated by a convolution, the theory of discrete transforms (that was originally developed for digital signal processing) indicates to us how to compute this operation quickly with some Fast Fourier Transforms. With this method, some primes with more than 10,000 and 100,000 digits were found.

Generalized Fermat Numbers are more numerous than Mersenne numbers at equal size and many of them are waiting to be discovered to fill the gaps between the Mersenne primes already found or not yet found. If you are interested in the search, you are welcome to the Generalized Fermat Prime Search, Fermatsearch and Search for GFN factors!

(Gallot, Y. "Generalized Fermat prime search."



Searching for Generalized Fermat Primes is not easy, but offers thrill and excitement.
Not all software for Fermat factors divisors can be used. Note that Fermat.exe and GMP-Factor are optimized for Fermat factors only, while PFGW can be used for both searches.

What is required in our context is a test of a total of 41 base pairs, which include "xGF"s for (a,b)=(3,2), (4,3), (5,2), (5,3), (5,4), (6,5), ... This is carried out by the program PFGW with the settings -gxo -a1 , where "-gx" generates the "default" set of 41 pairs of interest.

Proth.exe is only testing GFNs for bases (a,b)=(3,1), (5,1), (6,1), (10,1), (12,1), and so is not suitable for our extensive search.

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