Question 135
Find all pairs $(m, n)$ of integers that satisfy $m^2 + n^2 = 65$. Use this to prove the inequalities
$$
\frac{4 + 4\sqrt{10} + 6\sqrt{2}}{\sqrt{65}} \lt \pi \lt \frac{4}{11}(3 + \sqrt{2} + 2\sqrt{5}).
$$
$$
\frac{4 + 4\sqrt{10} + 6\sqrt{2}}{\sqrt{65}} \lt \pi \lt \frac{4}{11}(3 + \sqrt{2} + 2\sqrt{5}).
$$