Problem

Source: Thailand TSTST 2024 P3

Tags: number theory, primitive root



Recall that for an arbitrary prime $p$, we define a primitive root modulo $p$ as an integer $r$ for which the least positive integer $v$ such that $r^{v}\equiv 1\pmod{p}$ is $p-1$. Prove or disprove the following statement: For every prime $p>2023$, there exists positive integers $1\leqslant a<b<c<p$ such that $a,b$ and $c$ are primitive roots modulo $p$ but $abc$ is not a primitive root modulo $p$.