Elementary Number Theory

Abstract

This is a lecture about some classical topics in Elementary Number Theory, such as the distribution of primes, modular arithmetic, the quadratic reciprocity law, and representations of integers by quadratic forms.

Objective

The participants will learn about some very classical and foundational topics in number theory. By solving exercises and doing numerical experiments, the students will learn how to attack elementary number theoretical problems and make their own conjectures about properties of integers and prime numbers. A focus will be put on the group work and joint solution of exercise problems, which helps to foster the student's mathematical communication skills. The results presented in the lecture also lay the foundation for more advanced courses, such as Algebraic Number Theory.

Content

Elementary number theory is one of the oldest branches of mathematics, dating back at least to the ancient Greeks. The term "elementary" refers to the fact that the problems are simple to state, and the proofs only rely on elementary tools, e.g. basic algebra but no (or very little) calculus. However, "elementary" should not be confused with "easy". It turns out that many elementary problems are very hard to solve: Some of the most difficult and long-standing problems of modern mathematics originate from elementary number theory! In the course, we will cover the following classical topics: - Prime numbers: Fundamental Theorem of Arithmetic, Euclidean Algorithm, Bertrand's postulate, Prime Number Theorem - Number-theoretic functions: Euler's totient function, Dirichlet convolution, Moebius inversion, perfect numbers, Mersenne primes - Modular arithmetic: basic group theory, Chinese Remainder Theorem, Fermat's Little Theorem, Quadratic Reciprocity Law - Quadratic forms: Pythagorean triples, Congruent numbers, Pell's equation and continued fractions, sums of squares, reduction theory of binary quadratic forms, Gauss composition If time permits, we may either discuss some applications to cryptography, e.g. the RSA cryposystem, or give an overview of some more advanced topics in number theory, e.g. the Birch-Swinnerton-Dyer conjecture.

Resources