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I'm having a little trouble, understanding the necessity of the assumptions in Artin's Conjecture.

Artin's primitive root conjecture states, that:

for any $$a\in \mathbb{Z}\setminus \{-1\}$$ and $$a \neq b^2$$ There are infinitely many primes $p$ , such that $a$ is a primitive root modulo $p$.

Can anybody explain why these restrictions are necessary?

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The hypothesis is that $a$ is not a square, not merely that it is squarefree.

Suppose $a=b^2$.

If the order of $b$ is even, then the order of $a$ is half the order of $b$ and so $a$ cannot have the maximum order.

If the order of $b$ is odd, then the order of $a$ is equal to the order of $b$ and so $a$ cannot have the maximum order because the maximal order is $p-1$, which is even.

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  • $\begingroup$ Thank you for your answer $\endgroup$ – limeeattack May 16 at 9:33

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