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One knows that if $a$ is a primitive root modulo $p$, then $(\frac{a}{p})=-1$.
Thus, if $(\frac{a}{p})=1$ we know that $a$ is not a primitive root modulo $p$.

Are there any other properties between Legendre symbol and primitive roots?

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Suppose that $p-1$ is divisible by $3$. If $a$ is a cubic residue of $p$, then $a$ is not a primitive root of $p$. Similarly for other prime divisors $q$ of $p-1$.

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