Integral points on cubic twists of Mordell curves

Fix a non-square integer k ≠ 0 . We show that the number of curves E B : y 2 = x 3 + k B 2 containing an integral point, where B ranges over positive integers less than N , is bounded by ≪ k N ( log N ) - 1 2 + ϵ . In particular, this implies that the number of positive integers B ≤ N such that - 3 k B 2 is the discriminant of an elliptic curve over Q is o ( N ). The proof involves a discriminant-lowering procedure on integral binary cubic forms.