Integral points on the congruent number curve

Abstract. We study integral points on the quadratic twists ED : y2 = x3 − D2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2ED(Q) in ED(Q) and show that their total is (3.8)rank ED(Q). We further show that the average number of non-torsion integral points in this family is bounded above by 2. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations aX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d, has at most (3.6)ω(abcd) integer solutions.