Module Identifier | MA30810 | ||

Module Title | NUMBER THEORY | ||

Academic Year | 2000/2001 | ||

Co-ordinator | Dr T P McDonough | ||

Semester | Semester 2 | ||

Pre-Requisite | MA21410 | ||

Course delivery | Lecture | 19 x 1hour lectures | |

Seminars / Tutorials | 3 x 1hour example classes | ||

Assessment | Exam | 2 Hours (written examination) | 100% |

Resit assessment | 2 Hours (written examination) | 100% |

**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

**Reading Lists**

**Books**
**** Recommended Text**

G A Jones & J M Jones.
*Elementary Number Theory*. Springer
**** Supplementary Text**

I Niven, H S Zuckerman & H L Montgomery.
*An Introductiono the Theory of Numbers*. Wiley

G H Hardy & E M Wright.
*An Introduction to the Theory of Numbers*. 5th edition. OUP

H E Rose.
*A Course in Number Theory*. Oxford Science Publications

R B J T Allenby & E J Redfern.
*Introduction to Number Theory with Computing*. Arnold

H Rademacher.
*Lectures on Elementary Number Theory*. Blaisdell

I M Vinogradov.
*Elements of Number Theory*. Dover