On the distribution of congruent numbers (Q1959765)

A positive integer is said to be a congruent number if it is equal to the area of a right triangle with rational sides. It is known that a positive integer \(n\) is a congruent number if and only if the elliptic curve \(y^2=x(x^2-n^2)\) has positive Mordell-Weil rank. In this article the authors prove that for any integer \(m>1\), any congruence class modulo \(m\) contains infinitely many congruent numbers \(n\), inequivalent modulo squares, such that the rank of the elliptic curve \(y^2=x(x^2-n^2)\) is greater than or equal to two.