Number theory
Number theory is the field of mathematics associated with studying the integers.
Contents
Introductory Topics
The following topics make a good introduction to number theory.
- Primes
- Composite numbers
- Divisibility
- Division Theorem (the Division Algorithm)
- Base numbers
- Diophantine equations
- Modular arithmetic
Intermediate Topics
An intermediate level of study involves many of the topics of introductory number theory, but involves an infusion of mathematical problem solving as well as algebra.
Olympiad Topics
An Olympiad level of study involves familiarity with intermediate topics to a high level, a few new topics, and a highly developed proof writing ability.
Advanced Topics in Number Theory
Algebraic Number Theory
Algebraic number theory studies number theory from the perspective of abstract algebra. In particular, heavy use is made of ring theory and Galois theory. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of 
. Famous problems in algebraic number theory include the Birch and Swinnerson-Dyer Conjecture and Fermat's Last Theorem.
Analytic Number Theory
Analytic number theory studies number theory from the perspective of calculus, and in particular real analysis and complex analysis. The techniques of analysis and calculus are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the Riemann Hypothesis.
Elliptic Curves and Modular Forms
(I don't really feel like writing this right now. Any volunteers?)
Resources
Books
- Introductory
- the Art of Problem Solving Introduction to Number Theory by Mathew Crawford (details)
- General Interest
Miscellaneous
- Intermediate
- Olympiad
Other Topics of Interest
These are other topics that aren't particularly important for competitions and problem solving, but are good to know.
