holds for all relatively prime to , not just for primitive roots. So if , what do you get if you raise both side to the power ?
let p be a prime congruent to 1 mod 3
let a be a prim root of p
suppose a is not congruent to 0 mod p
show that there is no solution x for the congruence x^2 = a mod p
i know by some theorem about primitive roots of primes that x^2 - a = 0 mod p has at most 2 solutions
i also know that the order of a modulo p is phi(p) = p -1, and so a^(p - 1) = 1 mod p, and p - 1 = 3k for some k, since p = 1 mod 3.
anyone have any hints? i started also by noting that (x^2)^(p - 1) = a^(p -1) mod p, and tried factoring some stuff with that, that is after assuming for a contradiction that in fact there is a solution x