|Integer factorization is one of the most interesting
things in computational number theory. First of all it is closely related
to cryptography, that's why large networks spend months of CPU-time on
cracking crypto-keys and number theorists invent new factoring algorithms
trying to factor numbers of some kind. One of the oldest factoring projects
is famous Cunningham
Project which deals with numbers of the form bn±1,
b<13, up to large n's. Such methods as MPQS and NFS were found in attempt
to split some Cunningham composites.
A number of new factoring projects has been announced since those times. Each of them concerns numbers of some special kind, therefore some special factoring methods are involved. However, it should be noticed that these numbers have one common feature: their form is suitable for quick deterministic primality tests, e.g. N±1 tests. This happens due to such a tendency that at first people use some kind of numbers to find primes, but then, after finding (or not) some primes, people begin to factor composites.
This tendency also takes place in XYYXF project. Paul Leyland was first who started the search for primes of the form xy + yx, some people joined this search later. But numbers of the form xy + yx are not suitable for fast deterministic primality tests, they are not cyclotomic and may not be easily represented in another algebraic forms to make them factorable with known fast algorithms. At the same time, their factors sometimes have special form. This project coordinates people to improve factoring methods in different ways, or even to find some new algorithms...