How to find $\gcd(a^{2^m}+1,a^{2^n}+1)$ when $m \neq n$?
Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd