# Theorems on Primitive Roots

Let g be a primitive root modulo $p$($p$ is an odd prime) with $g^{p-1}\ncong{1}\ (mod\ p^{2})$. I am interested in proving that

$$g^{(p-1)p^{m-2}}\ncong{1}\ (mod\ p^{m})$$ for every $m\geq{2}$

So far, I have proven that $a^{p^{m}-p^{m-1}}\equiv{1}\ (mod\ p^{m})$ given that $gcd(a,p)=1$. If that might be of help...

• math.stackexchange.com/questions/332760/… – lab bhattacharjee Nov 2 '17 at 3:35
• @labbhattacharjee I'm sorry, but it is not immediately clear to me, how this is relevant – Gwen Di Nov 2 '17 at 3:48
• From the given condition, $g$ is a primitive root $\pmod{p^2}$ . It is sufficient to establish $g$ is a primitive root $\pmod{p^m}$ – lab bhattacharjee Nov 2 '17 at 3:59