Testing for primitive roots using quadratic non residue and Jacobi symbol

Question

Is this always true for all cases??

$a$ is a primitive root $modulo$ $n$ $⇒$ $\left(\dfrac{a}{n}\right) = -1$

Is the converse also always true?

$\left(\dfrac{a}{n}\right)$ $= -1$ $⇒$ $a$ is a primitive root modulo n.

So if I'm testing for a primitive root modulo n then all I have to do is just evaluate the Jacobi symbol?

Answer 1

For both questions, the answer is No.

For the first, note that $2$ is a primitive root $\pmod 9$ but $$\left(\frac 29\right)=\left(\frac 23\right)^2=(-1)^2=1$$

For the second, note that $-1$ is not a square $\pmod 7$ and neither is it a primitive root.