Question 1.
With $0 < a < p$, $p$ prime and $\gcd(a,p-1)=1$, is it true that $0, 1, 2^a, ...,(p-1)^a$ is a complete residue system modulo $p$? If not, will a similar statement hold?
Question 2.
I was told it works for $a = 3$, does anyone know a simple proof of it in this particular case?