  Fix a non-square integer $k\neq 0$. 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 $O_k(N(\log N)^{-\frac{1}{2}+\epsilon})$.
In particular, this implies that the number of positive integers $B\leq N$ 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.
