Excludent Factorization Method

Also known as the difference of squares. It was first used by Fermat and improved by Gauß. Gauss looked for Integers $x$ and $y$ satisfying

$$y^2\equiv x^2-N\ \left({{\rm mod\ } {E}}\right)$$

for various moduli $E$. This allowed the exclusion of many potential factors. This method works best when factors are of approximately the same size, so it is sometimes better to attempt $mN$ for some suitably chosen value of $m$.

See also Prime Factorization Algorithms