Product of two primitive roots $\bmod p$ cannot be a primitive root.

Question

I recently proved that the product of all primitive roots of an odd prime $p$ is $\pm 1$ as an exercise. As a result, I became interested in how few distinct primitive roots need to be multiplied to guarantee a non-primitive product. Testing some small numbers, I have the following claim: "If $n$ and $m$ are two distinct primitive roots of an odd prime $p$, then $nm \bmod p$ is not a primitive root."

I've tried to make some progress by rewriting one of the primitive roots as a power of the other, but haven't been able to see any argument which helps me prove the result. Any help would be appreciated.

Answer 1

A primitive root of an odd prime $p$ must be a quadratic non-residue of $p$, and the product of two non-residues is a residue.

Answer 2

Take any generator $g$ of the multiplicative group $\Bbb F_p^\times$. If $r$ and $s$ are primitive roots then $r=g^u$ and $s=g^v$ where $u$ and $v$ are coprime with $p-1$, so $u$ and $v$ are odd. Then, $rs=g^{u+v}$ and $u+v$ is even, so $rs$ is not a primitive root.

Esentially it is the same reasoning than André's.

Answer 3

1. If $d:=\gcd(k,p-1)>1$, then $n^k$ is not a primitive root modulo $p$.

   $(n^k)^{\frac{p-1}d} \equiv 1 \pmod p$ already

2. Supposed $m=n^k$is also a primitive root. Then by 1., $k$ must be odd, as $p-1$ is even.
3. But then $mn=n^{k+1}$ is not primitive root by 1., since $k+1$ is even.