Speeding Fermat's factoring method

A factoring method is presented which, heuristically, splits composite n in O(n1/4+ ) steps. There are two ideas: an integer approximation to √ (q/p) provides an O(n1/2+ ) algorithm in which n is represented as the difference of two rational squares; observing that if a prime m divides a square, then m2 divides that square, a heuristic speed-up to O(n1/4+ ) 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 itself.