  Legendre discovered that the continued fraction expansion of $\sqrt N$ having
odd period leads directly to an explicit representation of $N$ as the sum of
two squares. In this vein, it was recently observed that the continued fraction
expansion of $\sqrt N$ having even period directly produces a factor of
composite $N$. It is proved here that these apparently fortuitous occurrences
allow us to propose and apply a variation of Shanks' infrastructural method
which significantly reduces the asymptotic computational burden with respect to
currently used factoring techniques.
