A deterministic algorithm for solving n = fu2+gv2in coprime integers u and v

We give a deterministic algorithm for finding all primitive representations of a natural number n in the form fu2+gv2where f and g are given positive coprime integers, and n ≥ f + g+ 1, (n, fg) = 1. The running time of this algorithm is at most ≥ (n1 4(logn)3(loglogn)(logloglogn)), uniformly in and g