Question 38
Find all pairs of positive integers $(m, n)$, $m \gt n$, such that
$$
\left[m^2 + m n, m n - n^2\right] + [m - n, m n] = 2^{2005}
$$
where $[a, b]$ denotes the least common multiple of $a$ and $b$.
$$
\left[m^2 + m n, m n - n^2\right] + [m - n, m n] = 2^{2005}
$$
where $[a, b]$ denotes the least common multiple of $a$ and $b$.