TITLE: Finding a primitive root
QUESTION [7 upvotes]: I'm trying to find a small primitive root modulo $p^k$, where $p$ is prime.  My strategy is to test small numbers $g=2,3,\ldots$ until I find a primitive root modulo $p$.  That is, until $ord_pg=\phi(p)=p-1$.  There are results that suggest such a search won't take long.
To go from modulo $p$ to modulo $p^k$, there is the following well-known theorem:

Thm: If $g$ is a primitive root modulo $p$, then $$\begin{cases} g+p & g^{p-1}\equiv 1\pmod{p^2}\\g & g^{p-1}\not\equiv 1\pmod{p^2}\end{cases}$$
  is a  primitive root modulo $p^k$, for all $k\in\mathbb{N}.$

However, I was unable to find an example that falls into the first case (except $p=2$, trivially).  Hence, I have two related questions:

Are there any odd primes $p$, and primitive roots $g$, such that $g^{p-1}\equiv 1\pmod{p^2}$?
Are there any odd primes $p$, and minimal primitive roots $g$, such that $g^{p-1}\equiv 1\pmod{p^2}$?

REPLY [2 votes]: Surprisingly, only two odd primes are known such that the smallest primitive root mod $p$ is not a primitive root mod $p^2$: $40487$ and $6692367337$. See OEIS:A055578.