---
title: Quadratic Residues
description: Quadratic residues are integers that have a square root modulo a given prime. Learners will understand Euler's criterion, quadratic reciprocity, and how to solve quadratic congruences in number theory.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/quadratic-residues
---

# Quadratic Residues

Quadratic residues are integers that have a square root modulo a given prime. Learners will understand Euler's criterion, quadratic reciprocity, and how to solve quadratic congruences in number theory.

## Where to start

Start with the Art of Problem Solving Wiki, a free, detailed article on quadratic residues with examples. If you only use one resource, make it Neal Koblitz's A Course in Number Theory and Cryptography, a number theory textbook written with cryptography in mind.

## Available Resources

1 Books • 1 Courses • 3 Websites

## Websites

### 1. Brilliant.org

Brilliant's wiki page on quadratic residues, covering the definition, Euler's criterion, the Legendre symbol, and quadratic reciprocity, with worked problems. Readers learn to decide whether an integer is a square modulo a prime.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://brilliant.org/wiki/quadratic-residues/

**Tags:** quadratic-residues, legendre-symbol, quadratic-reciprocity, number-theory

### 2. Art of Problem Solving Wiki

Art of Problem Solving wiki entry on quadratic residues, defining them modulo n, introducing the Legendre symbol and Euler's criterion, and showing how residues are used in olympiad-style number theory problems. Good for competition-oriented learners building modular arithmetic skills.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://artofproblemsolving.com/wiki/index.php/Quadratic_residue

**Tags:** quadratic-residues, modular-arithmetic, legendre-symbol, competition-math, number-theory

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Number Theory

Four-week UC San Diego course building from division with remainder and modular arithmetic to Euclid's algorithm and RSA. Finishing it, you can encrypt and decrypt messages and break RSA implementations that reuse or mishandle keys.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/number-theory-cryptography

**Tags:** courses, mathematics-statistics, mathematics

## Books

### 1. A Course in Number Theory and Cryptography

**Author:** Neal Koblitz

Graduate Texts in Mathematics volume introducing elementary number theory alongside its cryptographic applications: finite fields, quadratic residues and reciprocity, public-key cryptosystems, primality testing, factoring, and elliptic curves. Readers learn the mathematics behind RSA and elliptic-curve cryptography.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0387942939?tag=edmonddante07-20

**Tags:** number-theory, cryptography, quadratic-residues, elliptic-curves, public-key-cryptography

---

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