General description
The theory of numbers is one of the oldest branches of mathematics. It is concerned with a study of the most basic objects of mathematics - the integers. The module leads to a study of the topics (i) solution of polynomial congruences, (ii) the quadratic reciprocity law, (iii) multiplicative arithmetic functions, e.g. the sum of the factors as an integer, (iv) representing numbers by forms, (only special quadratic forms are considered).

Aims
To provide an introduction to some topics in classical number theory.

Learning outcomes
On completion of this module, a student should be able to:

• examine consequences of the Euclidean algorithm and factorisation of integers into products of primes;
• solve polynomial congruences;
• derive consequences of quadratic residue law;
• establish identities involving multiplicative functions;
• provide formal proof of propositions.

Syllabus
1. Divisibility and Congruence
2. Classical Congruences
3. General Techniques for Solving Polynomial Congruences
4. Multiplicative Functions and Related Identities
5. Diophantine Equations
6. Equivalence of Quadratic Forms