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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?

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    Isn't this false for $a=2$ since $x^2 = (-x)^2$? Did you mean $\gcd(a,p-1)=1$?2011-09-14
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    If $p=7, 2^3=1 \pmod p$ so it fails2011-09-14
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    @Srivatsan Narayanan: yep you are right, i mistyped, how to correct my mistake? make another question?2011-09-14
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    Well, Pierre, what did you intend to be the question? I guess many of the answers cover both the $p$ and $p-1$ cases. Perhaps you can *add* the new question to the old question. But do not change the question substantially especially if that will invalidate the existing answers.2011-09-14
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    Also see [MSE question](http://math.stackexchange.com/questions/57162) re the number of equivalence classes of (in notation of present question) $k^a\mod p$2011-10-11

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