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$\underline{Theorem \ 1}$ : The group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ is cyclic and its order is $p-1$.

$\underline{Theorem \ 2}$ : Let $k\ge 1$ an integer and $p$ an odd prime number. Then $p^{k}$ admits a primitive root.

Thanks in advance !

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Citing Wikipedia:

Gauss defined primitive roots in Article 57 of the Disquisitiones Arithmeticae (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime n.

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  • $\begingroup$ Interesting. In number theory Gauß has more than $25$ theorems, unbelievable ! And what about the first one ? $\endgroup$ – Maman Apr 2 '17 at 15:19
  • $\begingroup$ @Maman, the first one is equivalent to the existence of primitive roots for primes. $\endgroup$ – lhf Apr 2 '17 at 15:21
  • $\begingroup$ Have you read his work on binary forms ? $\endgroup$ – Maman Apr 2 '17 at 15:25

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