Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers. [duplicate]
Solution 1:
Let $a_n = 2^{2^n}+1$. Then:
$$ a_{n+m} = (a_n-1)^{2^m}+1, $$ hence: $$ a_{n+m}\equiv (-1)^{2^m}+1 = 2\pmod{a_n}, $$ from which it follows that: $$ \gcd(a_{n+m},a_{n}) = \gcd(2,a_n)=1 $$ since $a_n$ is odd.