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.

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Persistent URL dx.doi.org/10.1090/S0025-5718-1990-1023762-3
Journal Mathematics of Computation
Hardy, K. (Kenneth), Muskat, J.B. (Joseph B.), & Williams, K.S. (1990). A deterministic algorithm for solving n = fu2+gv2in coprime integers u and v. Mathematics of Computation, 55(191), 327–343. doi:10.1090/S0025-5718-1990-1023762-3