# Math Help - A congruence problem

1. ## 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.

2. 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 $(a,p)=1$ this is the same as solving $(xa^{-1})^2\equiv-1\bmod{p}$ or in other words solving $y^2\equiv-1\mod{p}$.

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