Speeding Fermat's factoring method

A factoring method is presented which, heuristically, splits composite $n$ in $O(n^{1/4+\epsilon})$ steps. There are two ideas: an integer approximation to $\surd(q/p)$ provides an $O(n^{1/2+\epsilon})$ algorithm in which $n$ is represented as the difference of two rational squares; observing that if a prime $m$ divides a square, then $m^2$ divides that square, a heuristic speed-up to $O(n^{1/4+\epsilon})$ 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 $n$ itself.