--- title: Number Theory description: Number theory is the study of the properties and relationships of integers. You will understand concepts like prime factorization, divisibility, modular arithmetic, and cryptography, enabling you to solve complex mathematical puzzles and secure digital communications. category: mathematics subcategory: arithmetic-and-pre-algebra difficulty: beginner, intermediate, advanced url: /subject/number-theory --- # Number Theory Number theory is the study of the properties and relationships of integers. You will understand concepts like prime factorization, divisibility, modular arithmetic, and cryptography, enabling you to solve complex mathematical puzzles and secure digital communications. ## Where to start Start with A Friendly Introduction by Joseph H. Silverman, whose short chapters need no algebra beyond high school mathematics. If you only use one resource, make it An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, a classic proof-heavy text. For the cryptography side, the Number Theory course on Coursera from UC San Diego builds up to RSA. ## Available Resources 3 Books • 4 Courses • 5 Websites ## Websites ### 1. This Week's Finds in Mathematical Physics, Week 201 **Author:** John Baez John Baez's 2004 expository column introducing Galois theory through symmetry: Galois groups of field extensions, quadratic and cyclotomic number fields, the fundamental theorem, why the quintic has no radical solution, and classical construction problems, with pointers toward class field theory. **Difficulty:** Intermediate | **Price:** Free **Link:** https://math.ucr.edu/home/baez/week201.html **Tags:** galois-theory, field-extensions, galois-groups, algebraic-number-theory, cyclotomic-fields ### 2. Art of Problem Solving Number Theory Community-maintained wiki pages on divisibility, modular arithmetic, Diophantine equations and related olympiad topics, each paired with worked contest problems. Working through them builds the pattern recognition competition number theory questions demand. **Difficulty:** Beginner | **Price:** Free **Link:** https://artofproblemsolving.com/wiki/index.php/Number_theory **Tags:** competition-math, olympiad, divisibility, modular-arithmetic, problem-solving ### 3. numbertheory.org Numbertheory.org is a comprehensive online hub for learning number theory, offering curated tutorials, lecture notes, bibliographies, and a wide collection of links to papers, software, and related resources. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://numbertheory.org **Tags:** websites, mathematics-statistics, mathematics ### 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 ### 5. primes.utm.edu Primes.utm.edu, The Prime Pages, is a comprehensive number theory resource focused on primes, offering extensive prime catalogs (including Mersenne primes), historical notes, tables, and references to algorithms, software, and research data. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://t5k.org/ **Tags:** websites, mathematics-statistics, mathematics ## Courses ### 1. Elementary Number Theory Explore the beauty of numbers! This elementary number theory course covers divisibility, primes, congruences, and more. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/number-theory **Tags:** number-theory, divisibility, modular-arithmetic, primes ### 2. Introduction to Number Theory Explore the fascinating world of number theory! This course covers divisibility, congruences, and more, laying a foundation for advanced math. **Difficulty:** Beginner | **Price:** Paid **Link:** https://artofproblemsolving.com/school/course/ntonumbertheory **Tags:** competition-math, divisibility, modular-arithmetic, olympiad-prep ### 3. Pre-Algebra Master fractions with Sal Khan's Pre-Algebra course! Build a strong foundation for algebra and beyond. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/pre-algebra **Tags:** pre-algebra, fractions, ratios, negative-numbers, arithmetic ### 4. 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 ## Youtubes ### 1. Mathologer **Author:** Burkard Polster Deep mathematical explorations covering number theory, geometry, combinatorics, and recreational mathematics with rigorous proofs. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/@Mathologer **Tags:** youtube, mathematics, discrete-mathematics ## Books ### 1. An Introduction to the Theory of Numbers **Author:** Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery Classic Wiley text spanning divisibility, congruences, quadratic reciprocity, Diophantine equations, continued fractions, and analytic estimates on prime distribution. Dense and proof-heavy, it suits a reader who already has some comfort writing rigorous arguments. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0471641545?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, arithmetic--pre-algebra ### 2. Elementary Number Theory **Author:** David M. Burton One-semester undergraduate text presenting divisibility, congruences, Fermat's and Euler's theorems, primitive roots and quadratic reciprocity, with historical background on each result. Proofs are worked out in full detail, making it usable without an instructor. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0072325690?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, arithmetic--pre-algebra ### 3. A Friendly Introduction **Author:** Joseph H. Silverman Short chapters that lead the reader to discover patterns numerically before proving them, covering Pythagorean triples, congruences, Fermat's little theorem, quadratic reciprocity and RSA. Requires no algebra background beyond high school mathematics. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0132637995?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/number-theory