> After my latest miss-steps I went back, looked everything over again
> and finally noticed something that was bizarrely, simply obvious and
> staring me in the face.
>
> Here's the factoring algorithm that results.
>
> Given an odd target composite T, for a test odd prime p, iterate
> through non-zero integers k until
>
> T - k^2 mod p
>
> is a quadratic residue. If no k will work, discard that prime p and
> try another. If one does work then get á from
>
> á^2 = (T - k^2)(k^2)^{-1} mod p.
>
> Then you have residues of your factors modulo p from
>
> f_1 = ák mod p
>
> and
>
> f_2 = á^{-1}(1 + á^2)k mod p
>
> where f_1*f_2 = T.
>