Using primitive roots show that for any prime there's $n$ that the eqn $x^n \equiv x+1 \pmod{p}$ has an integer solution

Question

Using primitive roots show that for any prime $p > 3$ there exists $n \in {2, 3,..., p − 2}$ such that the polynomial equation $x^n \equiv x+1 \pmod{p}$ has an integer solution.

How do I do this?

Answer 1

If $x$ is a primitive root $\forall y\in\mathbb{Z}/p\mathbb{Z} $ $ \exists 1<\alpha<p$ such that $y=x^\alpha$.

So there is $n$ such that $x+1=x^n$