Can they both be perfect squares? Let $X$ and $Y$ be positive integers. Prove that at least one of $X^2+Y$ and $Y^2+X$ is not a perfect square.

If $X^2+Y=A^2$ then $Y\ge{2X+1}$ and thus $Y>{X}$.
If $Y^2+X=B^2$ then $X\ge{2Y+1}$ and thus $X>{Y}$.
And this is a contradiction.
