Baltic Way 1990 · Problem 13

Solution: For any solution $(m, n)$ of the equation we have $m^{2}-7 n^{2}=1$ and

$$1=\left(m^{2}-7 n^{2}\right)^{2}=\left(m^{2}+7 n^{2}\right)^{2}-7 \cdot(2 m n)^{2} .$$

Thus $\left(m^{2}+7 n^{2}, 2 m n\right)$ is also a solution. Therefore it is sufficient to note that the equation $x^{2}-7 y^{2}=1$ has at least one solution, for example $x=8, y=3$.