# Assumptions in Artin's primitive root conjecture

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?

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.