A possible defining characteristic of primitive roots.

Question

If $n$ is a primitive root $\bmod p$ ($p$ is an odd prime ) does there always exist a least residue $t$ such that $n^t \equiv t \pmod p$ ?

Answer 1

Recall that $n^{p-1}\equiv 1\pmod p$. For any $r\in\{0,\ldots,p-1\}$, find $a$ with $n^a\equiv r\pmod p$. By the Chinese Remainder Theorem, there exists $t\in\mathbb N_0$ with $t\equiv a\pmod{p-1}$ and $t\equiv r\pmod p$. For such $t$, we have $n^t\equiv t\pmod p$. As is the case for any nonempty set of $\mathbb N_0$, there certainly is a minimal such $t$.