Abstract
We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These numbers are called θ-congruent. We give an elliptic curve criterion for determining whether a given integer n is θ-congruent. We then consider the "density" of integers n which are θ-congruent, as well as the related problem giving the "density" of angles θ for which a fixed n is congruent. Assuming the Shafarevich–Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.
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