The following facts about primitive roots of an odd prime seem to be well known. For example, they both appear as exercises in Burton's Elementary Number Theory.
Let $p$ be an odd prime. Then:
(1) Any primitive root of $p^2$ is a primitive root of $p^k$ for every positive integer $k$.
(2) Any odd primitive root of $p^k$ is a primitive root of $2p^k$.
I thought these facts might be from Gauss' Disquisitiones Arithmeticae, but I couldn't find them there. Does anyone know the origin of these two facts?