The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves epsilon(D): y(2) = x(3) -D(2)x, where D ranges over positive squarefree integers less than N , is O ( N (log N ) (- 1/4 +E )). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of HeathBrown's method on estimating the average size of the 2-Selmer groups of the curves in this family.<br /> (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).