I want to show that if $3$ is a primitive root modulo $p$ if $p$ is a prime of the form $2^n+1$ for some $n>1$.
First, I wrote $3^{p-1} \equiv 1 \mod p$. Then writing it as $(1+2)^{p-1}$, we see that $p| 2+ \dots 2^{p-1} =2^p - 2$ and I stuck at this point.
Also, by a theorem we can see that $3 $ is a quadratic residue modulo $p$. I wrote $3 \equiv n^2 \mod p$ but that is not helpful too.
So, how can I prove this theorem?