TITLE: Why 4 is not a primitive root modulo p for any prime p?
QUESTION [6 upvotes]: 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 !

REPLY [2 votes]: For any relevant $p$, the order of 2 will be larger than the order of $4 = 2\cdot 2$.