If r is a primitive root of p^2, p being an odd prime. show that the solutions of the congruence X^(p-1)=1 (mod p^2) are precisely the integer r^p, r^(2p),......r^((p-1)p)
If r is a primitive root of p^2, p being an odd prime. show that the solutions of the congruence X^(p-1)=1 (mod p^2) are precisely the integer r^p, r^(2p),......r^((p-1)p)
Check that all of these are solutions by substituion and that for .
Now if is solution then and so . Therefore, .
Now so there are solutions to this congruence. Which means we found all of them.