---
title: Diophantine Equations
description: This subject focuses on polynomial equations for which only integer or rational solutions are sought. Learners will understand classical methods for finding solutions, analyze solvability, and explore the connections between geometry and number theory.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/diophantine-equations
---

# Diophantine Equations

This subject focuses on polynomial equations for which only integer or rational solutions are sought. Learners will understand classical methods for finding solutions, analyze solvability, and explore the connections between geometry and number theory.

## Available Resources

2 Books • 1 Courses • 4 Websites

## Websites

### 1. MathWorld - Diophantine Equation

Wolfram MathWorld's reference entry on Diophantine equations, defining them, surveying linear, Pell, and higher-degree cases, and linking to related results such as Hilbert's tenth problem, with citations to the primary literature for readers who want to follow up on specific equations.

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

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

**Tags:** diophantine-equations, number-theory, pell-equation, hilberts-tenth-problem, integer-solutions

### 2. Brilliant.org - Diophantine Equations

Brilliant's wiki page on Diophantine equations, explaining linear equations solved with the Euclidean algorithm, Pythagorean triples, and Pell's equation, then using modular arithmetic and factoring to rule out or find integer solutions through worked examples and practice problems.

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

**Link:** https://brilliant.org/wiki/diophantine-equations/

**Tags:** diophantine-equations, number-theory, euclidean-algorithm, modular-arithmetic, pell-equation

### 3. Art of Problem Solving Wiki - Diophantine Equations

Art of Problem Solving's competition-oriented wiki article on Diophantine equations, covering linear equations, Pythagorean triples, Pell equations, and Fermat's Last Theorem, with methods such as modular restrictions and factoring illustrated through contest problems that build olympiad-style solving technique.

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

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

**Tags:** diophantine-equations, number-theory, math-competitions, olympiad-math, pell-equation

### 4. 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

## Books

### 1. Rational Points on Elliptic Curves (2nd edition)

**Author:** Joseph H. Silverman, John T. Tate

Undergraduate treatment of elliptic curves as Diophantine objects: the group law, Nagell-Lutz, Mordell's theorem, curves over finite fields, integer points, complex multiplication and cryptography. The author's page carries the table of contents and 2024-updated errata.

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

**Link:** https://www.amazon.com/dp/3319307576?tag=edmonddante07-20

**Tags:** elliptic-curves, diophantine-equations, number-theory, mordell-theorem, arithmetic-geometry

### 2. An Introduction to Diophantine Equations

**Author:** Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

A problem-based introduction to Diophantine equations covering elementary solving methods (factoring, modular arithmetic, inequalities, parametrization, infinite descent) and classical equations including Pell's and Pythagorean triples, with fully worked solutions aimed at olympiad and Putnam competitors and undergraduates.

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

**Link:** https://www.amazon.com/dp/0817645489?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, number-theory

## 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

---

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