Heron Triangles via Elliptic Curves

Abstract

Given a positive integer n, one may ask if there is a right triangle with rational sides having area n. Such integers are called congruent numbers, and are closely related to elliptic curves of the form y2 = x3−n2 x. In this paper, we generalize this idea and show that there is a correspondence between positive integers n associated with arbitrary triangles with rational sides having area n and the family of elliptic curves y2 = x(x− n τ)(x + n τ−1) for nonzero rational τ .