Speeding Fermat’s factoring method

Abstract

A factoring method is presented which, heuristically, splits composite n n in O ( n 1 / 4 + ϵ ) O(n^{1/4+\epsilon }) steps. There are two ideas: an integer approximation to √ ( q / p ) \surd (q/p) provides an O ( n 1 / 2 + ϵ ) O(n^{1/2+\epsilon }) algorithm in which n n is represented as the difference of two rational squares; observing that if a prime m m divides a square, then m 2 m^2 divides that square, a heuristic speed-up to O ( n 1 / 4 + ϵ ) O(n^{1/4+\epsilon }) steps is achieved. The method is well-suited for use with small computers: the storage required is negligible, and one never needs to work with numbers larger than n n itself.