Is there a way to find all integer solutions to a hyperbola equation? If it helps, I am specifically looking at "square" hyperbolas (i.e. of the form $\frac{x^2}{z} - \frac{y^2}{z}=1$), where z is an integer (although $\sqrt{z}$ is not necessarily rational). I suspect that there are a finite number of these solutions, but have no idea how I would go about showing that.
I want to be able to find integer solutions $(x,y)$ for a fixed $z$.
$$x^{2}-y^{2}=z \implies (x-y)(x+y)=z$$
So to find integer solutions, consider the factors of $z$.