Number theory

Number theory is the field of mathematics associated with studying the properties and identities of integers.

Overview

Number theory is a broad topic, and may cover many diverse subtopics, such as:

• Modular arithmetic
• Prime numbers

Some branches of number theory may only deal with a certain subset of the real numbers, such as integers, positive numbers, natural numbers, rational numbers, etc. Some algebraic topics such as Diophantine equations as well as some theorems concerning integer manipulation (like the Chicken McNugget Theorem ) are sometimes considered number theory.

Student Guides to Number Theory

• Introductory topics in number theory
  • Covers different kinds of integers such as prime numbers, composite numbers, and their relationships (multiples, divisors, and more). Also includes base numbers and modular arithmetic.
• Intermediate topics in number theory
• Olympiad topics in number theory
• Advanced topics in number theory

Resources

Books

• Introductory
  • the Art of Problem Solving Introduction to Number Theory by Mathew Crawford (details)
  • Elementary Number Theory: A Problem Oriented Approach by Joe Roberts (details) Out of print but if you can find it in a library or used, you might love it and learn a lot. Writen caligraphically by the author.
• General Interest
  • Fermat's Enigma by Simon Singh (details)
  • Music of the Primes by Marcus du Sautoy (details)
  • 104 Number Theory Problems by Titu Andreescu, Dorin Andrica, Zuming Feng

E-Book

• Problems in Elementary Number Theory by Hojoo Lee
• Intermediate Number Theory by Justin Stevens
• Number Theory by Naoki Sato

Online Courses

• Introductory Number Theory
  • Introduction to Number Theory
• Intermediate
  • Intermediate Number Theory

Other Topics of Interest

These are other topics that aren't particularly important for competitions and problem solving, but are good to know.

• Fermat's Last Theorem

Famous Unsolved Number Theory Problems

• Birch and Swinnerton-Dyer conjecture
• Collatz Problem
• Goldbach Conjecture
• Riemann Hypothesis
• Twin Prime Conjecture