The University of Southampton
Courses

# MATH1001 Number Theory

## Module Overview

The aim of this module is to introduce students to some of the basic ideas of number theory, and to use this as a context in which to discuss the development of mathematics through examples, conjectures, theorems, proofs and applications. The module will introduce and illustrate different methods of proof in the context of elementary number theory, and will apply some basic techniques of number theory to cryptography. One of the pre-requisites for MATH3078

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• analyse hypotheses and conclusions of mathematical statements
• apply different methods of proof to verify mathematical assertions, including proof by induction, by contrapositive and by contradiction
• solve systems of Diophantine equations using the Chinese Remainder Theorem & the Euclidean algorithm
• understand the basics of modular arithmetic
• state and prove Fermat's Little Theorem & its generalisation using Euler's function & use them to implement the RSA cipher & dicrete log cipher

### Syllabus

• Proof and Mathematical Logic • Number Theory • Divisibility, least common multiples, Euclid's algorithm. • Integer solutions of ax + by = c. • Prime numbers and prime-power factorisations, irrational numbers. • Existence of infinitely many primes. • Modular arithmetic, linear congruences. Chinese Remainder Theorem. • Fermat's Little Theorem. • Units, Euler's function, Euler’s Theorem. • Cryptography • Diffie-Hellman-Merkle and Rivest-Shamir-Adleman key exchange systems • Pythagorean triples & Fermat’s Last Theorem.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, workshops, private study

TypeHours
Follow-up work24
Supervised time in studio/workshop6
Lecture36
Tutorial12
Revision30
Preparation for scheduled sessions12
Total study time150

ROSEN K H (1988). Number Theory and its Applications.

JONES G A & JONES J M (1998). Elementary Number Theory..

### Assessment

#### Summative

MethodPercentage contribution
Coursework 40%
Written assessment 60%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.