If $p$ is an odd prime with primitive root $1

Question

I use excel computed till $p=23$, it's true. But is this always true? if not, could you pls give a counter example?

Answer 1

just to dot it that $r=14$ is a primitive root for $p=29$, but not for $p^2=841$. Instead $r+p = 43$ is a primitive root for $841$.