Quadratic residues and non-residues of p=4k+3 are complementary

Question

Given a prime p=4k+3 and an integer X, is it the case that if X is a quadratic residue of p, then -X (i.e. p-X) is NOT a quadratic residue of p?

How to prove this? (Kudos for a simple proof!)

Answer 1

We will use the notation of Legendre symbols. Suppose $x$ is a quadratic residue mod $p$ where $p \equiv 3 \mod 4$. Then $$\left(\frac{-x}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{x}{p}\right) = \left(\frac{-1}{p}\right)=-1.$$ The first equality follows from multiplicativity of the Legendre symbol. Then second equality follows from $x$ being a quadratic residue modulo $p$. And the last equality follows because $$\left(\frac{-1}{p}\right )= (-1)^{\frac{p-1}{2}} = (-1)^{\frac{4k+3-1}{2}} = (-1)^{2k+1} = -1. $$ Thus, $-x$ is not a quadratic residue modulo $p$.