On finding Fermat’s pairs

Factorization of hard integers is a known hard problem and improving existing methods is difficult. The paper investigates one approach of improving the first step of factorization algorithms looking for a Fermat pair. It is seen that directly searching for a Fermat pair is faster than is often assumed, but not fast enough to challenge factorization methods using a prime basis. Possibilities of improving the Quadratic Sieve and the General Number Field Sieve are investigated. The paper concludes that the only way to improve these algorithms essentially is to create a method that produces relations from a presentation of a suitable group.