Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$
Source: Indian Postal Coaching 2009 set 4 p6
Tags: diophantine, number theory, inequalities
Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$