√D and Pell Equations

Suppose we want to find a rational approximation for $\sqrt{2}$. Suppose that $$\sqrt{2}=\frac{p}{q}.$$ We can rearrange this to get $$p=\sqrt{2}q.$$ Squaring and rearranging again then yields $$p^2-2q^2=0.$$ However, recall that $x^2=2$ has no rational solution, so $x=\pm\sqrt{2}$ has no rational solution either. So we know $p^2-2q^2=0$ cannot have a solution in integers, or else we have a contradiction.

So, instead, let's try to solve $p^2-2q^2=1$, which would allow for a close enough approximation. This is know as a pell equation. A general pell equation would be of the form $$p^2-nq^2=1,$$ where $n$ is not a square.

To learn how to solve this equation, go here.

Explain a way to generate all integer solutions such that $x^2-2y^2=1$ or $x^2-2y^2=-1$.