Synopsis for Number Theory

Overview

Number theory is one of the oldest parts of mathematics. For well over two thousand years it has attracted professional and amateur mathematicians alike. Although notoriously `pure' it has turned out to have more and more applications as new subjects and new technologies have developed. Our aim in this course is to introduce students to some classical and important basic ideas of the subject.

Synopsis

The ring of integers; congruences; ring of integers modulo ; the Chinese Remainder Theorem. [2 lectures]

Wilson's Theorem; Fermat's Little Theorem for prime modulus; Euler's phi-function. Euler's generalisation of Fermat's Little Theorem to arbitrary modulus; primitive roots. [2 lectures]

Quadratic residues modulo primes. Quadratic reciprocity. [2 lectures]

Factorisation of large integers; basic version of the RSA encryption method. [2 lectures]

1. Alan Baker, A Concise Introduction to the Theory of Numbers (Cambridge University Press, 1984) ISBN: 0521286549 Chapters 1,3,4.
2. David Burton, Elementary Number Theory (McGraw-Hill, 2001).
3. Dominic Welsh, Codes and Cryptography, (Oxford University Press, 1988), ISBN 0-19853-287-3. Chapter 11.