# ASO: Number Theory - Material for the year 2019-2020

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8 lectures

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.

Students will learn some of the foundational results in the theory of numbers due to mathematicians such as Fermat, Euler and Gauss. They will also study a modern application of this ancient part of mathematics.

The ring of integers; congruences; ring of integers modulo $n$; the Chinese Remainder Theorem.

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.

Quadratic residues modulo primes. Quadratic reciprocity.

Factorisation of large integers; basic version of the RSA encryption method.

1) H. Davenport, *The Higher Arithmetic* (Cambridge University Press, 1992) ISBN 0521422272

2) G.H. Hardy and E.M. Wright, *An Introduction to the Theory of Numbers* (OUP, 1980) ISBN 0198531710

3) P. Erdős and J. Surányi, *Topics in the Theory of Numbers* (Springer, 2003) ISBN 0387953205

All these books contain some elementary material but go way beyond what is in the course. Full printed notes will be provided.