LECTURE NOTES OF WILLIAM CHEN

 

ELEMENTARY NUMBER THEORY

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.

To read the notes, click the links below for connection to the appropriate PDF files.

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 >>

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

Chapter 2 : ARITHMETIC FUNCTIONS >>

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

Chapter 3 : CONGRUENCES >>

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

Chapter 4 : QUADRATIC RESIDUES >>

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

Chapter 5 : SUMS OF INTEGER SQUARES >>

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

Chapter 6 : ELEMENTARY PRIME NUMBER THEORY >>

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

Chapter 7 : GAUSS SUMS AND QUADRATIC RECIPROCITY >>

• Gauss Sums
• Convergence of Fourier Series
• Proof of Theorem 7C