Elementary Number Theory

by WWL Chen

This set of notes has been used between 1981 and 1990 by the author at Imperial College, University of London.

The material has been organized in such a way to create a single volume suitable for an introduction to the elementary techniques of number theory. These are techniques that do not involve anything deep in algebra or analysis.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

Chapter 1: DIVISION AND FACTORIZATION (last uploaded on 30 January 2003)

Division
Factorization
Some Elementary Properties of Primes
Some Results and Problems Concerning Primes

Chapter 2: ARITHMETIC FUNCTIONS (last uploaded on 30 January 2003)

Introduction
The Divisor Function
The Moebius Function
The Euler Function
Dirichlet Convolution

Chapter 3: CONGRUENCES (last uploaded on 30 January 2003)

Introduction
Sets of Residues
Some Interesting Congruences
Some Linear Congruences
Some Polynomial Congruences
Primitive Roots
A Theorem of Gauss

Chapter 4: QUADRATIC RESIDUES (last uploaded on 30 January 2003)

Introduction
The Legendre Symbol
Quadratic Reciprocity
The Jacobi Symbol
The Distribution of Quadratic Residues

Chapter 5: SUMS OF INTEGER SQUARES (last uploaded on 30 January 2003)

Sums of Two Squares
Number of Representations
Sums of Four Squares
Sums of Three Squares

Chapter 6: ELEMENTARY PRIME NUMBER THEORY (last uploaded on 30 January 2003)

Euclid's Theorem Revisited
The Von Mangoldt Function
Tchebycheff's Theorem
Some Results of Mertens

Chapter 7: GAUSS SUMS AND QUADRATIC RECIPROCITY (last uploaded on 30 January 2003)

Gauss Sums
Convergence of Fourier Series
Proof of Theorem 7C