Euler’s Criterion for Quadratic Residues

Here we would show a beautiful application of Fermat’s Little Theorem.

An integer is a quadratic residue with respect to prime if for some integer .

Given a prime and an integer , how fast can we decide whether is a quadratic residue modulo ? As it turns out (courtesy to Euler), pretty fast.

Fermat’s Little Theorem

Introduction

This is a beautiful theorem which states that the difference between a positive integer and a prime power of , , is divisible by . It’s history can be found in Wikipedia: Fermat posited the theorem (and as usual, did not give the proof) while Euler first published a proof using mathematical induction. Below, we will state the theorem and provide a simple-to-understand proof using only modular arithmetic, followed by another simple proof using mathematical induction.