The congruent number problem and elliptic curves

Abstract

The Birch and Swinnerton-Dyer (BSD) conjecture and Goldfeld conjecture are fundamental problems in the arithmetic of elliptic curves. The congruent number problem (CNP) is one of the oldest problems in number theory which is, for each integer $n$, to find all the rational right triangles of area $n$. It is equivalent to finding all rational points on the elliptic curve $E^{(n)}:n{y}^2=x^3-x$. The BSD conjecture for $E^{(n)}$ solves CNP, and Goldfeld conjecture for this elliptic curve family solves CNP for integers with probability one. In this article, we introduce some recent progress on these conjectures and problems.