Is $n^{(p-1)/2}\equiv -1\pmod p$ implies $n$ is a primitive root modulo $p$?

Question

Let $p$ be an odd prime, $n$ be any integer, if $$n^{(p-1)/2}\equiv -1\pmod p,$$ is it always true that $k=p-1$ is the smallest positive integer satisfy $$n^k\equiv 1\pmod p?$$ This is the little lemma I need in my solution to a bigger problem. I first hold a skeptical mind that this is false, so I want to find is it possible $$n^{(p-1)/2}\equiv -1\pmod p\quad \text{and} \quad n^{(p-1)/3}\equiv1\pmod p?$$

I think this is not obvious, because $(p-1)/2$ does not have any relation with $(p-1)/3$, I have found an example, $7^6\equiv 1\pmod {19}$, but then I checked that $7^9\equiv 1\pmod {19}$ is also true.

Is there any counterexample? Or any proof?

Answer 1

No. Take $n=5$, $p=13$. Then $\frac{p-1}{2}=6$, $5^2=-1$ mod $p$, hence $5^6=-1$ mod $p$. However, $5^4=1$.

Answer 2

This is the little lemma I need in my solution to a bigger problem. I first hold a skeptical mind that this is false, so I want to find is it possible $$n^{(p-1)/2}\equiv -1\pmod p\quad \text{and} \quad n^{(p-1)/3}\equiv1\pmod p?$$

Yes, it is possible, e.g. $\bmod p\!=\!7\!:\,\ n\equiv -1\,\iff n^{\large 3}\equiv -1\,$ and $\ n^{\large 2}\equiv 1$

Generally: $\ \, a\equiv n^{\large (p-1)/6}\equiv -1\iff \,a^{\large 3}\!\equiv n^{\large (p-1)/2}\!\equiv -1\,$ and $\ a^{\large 2}\!\equiv n^{\large (p-1)/3}\!\equiv 1$