If $p$ is an odd prime with $(p - 1)/2$ primitive roots, is $p$ a Fermat prime?

Question

If $p$ is an odd prime and there are $(p - 1)/2$ primitive roots modulo $p$, then is $p = 2^k + 1$ for some nonnegative integer $k$?

This is the converse of a statement that I have already proved.

Answer 1

Yes. There are exactly $\varphi(p-1)$ primitive roots modulo $p$. If $\varphi(p-1)=\frac{p-1}2$, then $p-1$ has to be a power of $2$.