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

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.
