I am learning elementary number theory and with some reasons I have to find out all the elements with some specific oder in $Z_p^*$, which is just the multiplicative group induced by $Z_p$.
For example, given a prime $p=7$, I need to find out all $a\in \{1,2,..,6\}$ s.t. $a^3\equiv 1$ (mod $7$). This problem is equivalent to find out all $a\in Z_7^*$ with order of $3$.
We know $Z_p^*$ is also a cyclic group. So if we can find out all generators of the multiplicative group $Z_7^*$, say $g_1,g_2,...,g_k,$ then $(g_1)^2,(g_2)^2,...,(g_k)^2$ are all the elements with order of $3$ in $Z_7^*$.
But usually it is not easy to find out all the generators of $Z_p^*$, which are called primitive roots in number theory by the way.
So my problem is that, is there a quick method to find out all such primitive roots given any prime $p$, or is there any better way to find out all the elements with some specific oder in $Z_p^*$?
I don't remember too much group theory, so a hint or answer without involving deep group theory will be better. But any help will be greatly appreciated.
