On the number of coprime solutions of $y^2 = x^3 + k$

Abstract:

Let $N’(k)$ denote the number of coprime integer solutions $x,y$ of ${y^2} = {x^3} + k$. It is shown that $\lim {\sup _{k \to \infty }}N’(k) \geq 8$ and that $\lim {\sup _{k \to - \infty }}N’(k) \geq 12$.