Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m + 1,2^n + 1))}{\gcd(\phi(2^m + 1),\phi(2^n + 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. Nanang Susyanto, Jogjakarta
Problem
Source: Indonesia IMO 2010 TST, Stage 1, Test 4, Problem 4
Tags: number theory, greatest common divisor, inequalities, number theory proposed