I wanted to prove that when $n<\frac{p-1}{2}$ for prime $p$, then $x^n$ takes on more than or less than, but not equal to $3$ residues modulo $p$, for $x$ over the integers.
Obviously when $p\mid x$, the residue is $0$, and when $x=1$ the residue is $1$. However I'm not sure how to prove that other residues exist.