Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.
Source: Turkey Team Selection Test 2017 P6
Tags: number theory
Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.