Abstract
We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(logN)−14+ϵ). 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.
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