The average number of integral points on the congruent number curves

Abstract

We show that the total number of non-torsion integral points on the elliptic curves $E_D:y^2=x^3-D^{2}x$, where D ranges over positive squarefree integers less than N, is $O\left(N{(\log N)}^{-\frac{1}{4}+ϵ}\right)$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.