Thread: A congruence problem

Prove that x^2 = -a^2 mod p does not have a solution for p a prime = 3 mod4 and a = any integer not divisible by p (or prove that it has a solution if and only if p is a prime 1 mod 4). Please don't use the legendre symbol because I haven't learned that yet.

Originally Posted by machack

Prove that x^2 = -a^2 mod p does not have a solution for p a prime = 3 mod4 and a = any integer not divisible by p (or prove that it has a solution if and only if p is a prime 1 mod 4). Please don't use the legendre symbol because I haven't learned that yet.

Since this is the same as solving or in other words solving .

I know you don't know about Legendre symbols yet, but do you know how to solve this?