Suppose $\displaystyle p$ is prime and $\displaystyle n$ is a positive integer with $\displaystyle n<p$. Then if $\displaystyle n$ is a primitive root modulo $\displaystyle p$ and modulo $\displaystyle p^2$, then $\displaystyle n$ is a primitive root modulo $\displaystyle p^k$ for all positive integers $\displaystyle k$. I am sure I have seen this somewhere, but I can't for the life of me remember where. If I can't find it (and no one here knows its name), I will try to prove it (or disprove it if I am remembering it incorrectly).