Why 4 is not a primitive root modulo p for any prime p? I wonder why 4 is not a primitive root for any prime p ?
I've been trying to find an answer with no success so far. Any suggestion would be very helpful,
thanks in advance ! 

As $2^2=4,$
$4^{\left(\frac{p-1}2\right)}=2^{p-1}\equiv1\pmod p$ for any prime $p>2$ using Fermat's Little Theorem
So, the ord$_p4\le \frac{p-1}2<p-1$
