Recall that, if $p$ is an odd prime, a primitive root modulo $p$ is a generator of the cyclic (multiplicative) group $(\mathbb{Z} / p \mathbb{Z})^{\times}$. Let $p$ be an odd prime of the form $2^{2^{n}}+1$; show that $a$ is a primitive root $\bmod p$ if and only if $a$ is not a quadratic residue mod $p$. Use this result to prove that 7 is a primitive root modulo every such prime.