During the 17^{th} century, Pierre de Fermat and
Marin Mersenne studied two particular forms of numbers,
thinking that they could produce a large amount of prime
numbers or even to be ever prime. Mersenne communicated a
list of the primes of the form 2^{n}-1,
for all *n* < 257. It required many years of
labour to produce a correct version of the list, which is
close to Mersenne's list. In his correspondence with
Frénicle, Fermat expressed his conviction that if
*n* is a power of 2, then 2^{n}+1
is a prime. Fermat knew that 3, 5, 17, 257 and 65537 are
primes but later Euler showed that Fermat's conjecture is
false by discovering a factor to the next number.

In honour of the inspired pioneers, the numbers of the form
2^{n}-1 are now called the Mersenne
numbers and the numbers of the form
2^{n}+1 the Fermat numbers. The search for
Mersenne and Fermat primes has been greatly extended since
the 17^{th} century. Today, all the Mersenne primes
having less than 2,000,000 digits are known and all the
Fermat primes up to 2,000,000,000 digits! It was possible
because during the 19^{th} century some efficient
tests were discovered to check the primality of these
numbers. But at the same time, some tests as fast were also
found to test check the primality of all the numbers
*N* where the factorization of *N*-1 or
*N*+1 is known. Then many forms could be used to
find the largest known prime but surprisingly the search
was almost restricted to the Mersenne numbers. The famous
exceptions were (2^{148}+1)/17 (identified in 1951
by A. Ferrier by using hand computing method),
180.(2^{127}-1)^{2}+1 (discovered in 1951
by Miller and Wheeler) and 391581.2^{216193}-1
(found by the "Amdahl 6" in 1989).

In 1998, Y. Gallot remarked that the Discrete Weighted
Transform is a polynomial operation and that if the
representation of the numbers is not limited to the base 2,
then many numbers can be tested at the same speed as a
Mersenne number: the Generalized Fermat Numbers which are
the numbers of the form b^{n}+1, where
*n* is a power of 2. He implemented the
algorithm in 1999 in *Proth.exe* and since, he has
been optimizing. The theoretical hypothesis is now a
reality: the search for Generalized Fermat primes is
as fast as the search for a Mersenne primes of the same
size. With few tens of computers, many primes having
more than 100,000 digits were quickly found, the largest of
them has more than 800,000 digits. In 2002, P. Carmody
together with B. Frey made great strides in a Generalized
Fermat Number sieving algorithm. P.
Carmody is organizing a massive sieving
effort, with a powerful program written by D. Underbakke,
that even speeds up the Generalized Fermat Numbers
Search.

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 for the primes of the 21^{st} century,
you are welcome to the Generalized Fermat Prime Search!