Integral points on cubic twists of Mordell curves

Abstract

Fix a non-square integer $$k\ne 0$$ . We show that the number of curves $$E_B:y^2=x^3+kB^2$$ containing an integral point, where B ranges over positive integers less than N, is bounded by $$\ll _k N(\log N)^{-\frac{1}{2}+\epsilon }$$ . In particular, this implies that the number of positive integers $$B\le N$$ such that $$-3kB^2$$ is the discriminant of an elliptic curve over $$\mathbb {Q}$$ is o(N). The proof involves a discriminant-lowering procedure on integral binary cubic forms.