Integer solutions of a cubic equation

Question

With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$
I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into $x(x^2-y^2)=N-1$ and then tried to link it to Pell's Equation but so far I've got nothing, I don't even understand how Pell's Equation is solved (even though I know the method, I cannot grasp the intuition or reasoning behind it), I would appreciate any help.

Answer 1

You could always solve for $y$:$$y=\pm\sqrt{\frac{N-1-x^3}{-x}}$$ From here, you could use brute force to find some integer $x$ value that returns integer $y$ value.

You could always do it backwards as well, solving for $x$.

This is more difficult, as you are trying to solve a cubic equation. However, there is a cubic formula with much rigorous derivation behind it.

It can be found here.

Sadly, you would still have to use brute force methods. But the cube roots and such will make finding a $y$ value that will work much narrower.