$\Big [ \frac{p-1}{p}\Big ]_n$

Question

this is equivalent to solving $x^n\equiv-1\pmod p$ where $p$ is a prime number.
if $n=2$, the solution is iff $p\equiv 3 \pmod 4$.(if $p=2$ then all numbers are residues.)
I'm unsure about the rest of the powers.
if $p=2$ then all numbers are residues(thus $1$ is too).
if $p=3$ then iff $n\equiv 1 \pmod 2$
if $p=4$ then iff $n\equiv 1 \pmod 2$
can any of you find a general solution or help?