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...
