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Let $p>3$ be prime number and $a$ primitive root modulo $p^2$. Prove that $x^{p-1}\equiv 1 \pmod{p^2}$ solutions are $\overline{a}^p,\overline{a}^{2p},\ldots ,\overline{a}^{(p-1)p}$.

Any ideas on how to approach this problem?

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Hints

Because $a$ is a primitive root modulo $p^2$ you can conclude two things:

  • First every invertible element is of the form $a^i$ for some $0\leq i\leq p^2-p-1$ this allows you to take $x=a^i$ in order to solve the equation.
  • Second $a^t=1\mod p^2$ if and only if $t$ is divisible by $p^2-p$

Now use this two hints to prove that the solutions are as as you stated!

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