Integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer.

Question

Can we generate some parametric family of integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer.

I don't even know if there are finite or infinite number of solutions. For $k=7$, one of the solutions is $x=1,y=2$. I think will be only finite number of solutions to $7x^2=y^3-1$.

Answer 1

For each positive integer $k$, $k x^2 - y^3 + 1$ defines an elliptic curve. By Siegel's theorem, it has only finitely many integer solutions.