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.

### 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

• 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. • Pythagorean triples & Fermat’s Last Theorem. • Cryptography • Substitution ciphers, letter-frequency analysis. • Vigenere cipher, Babbage-Kasiski deciphering. • Diffie-Hellman-Merkle and Rivest-Shamir-Adleman key exchange systems

#### Special Features

For features such as field trips, information should be included as to how students with special needs will be enabled to benefit from this or an equivalent experience.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, workshops, private study

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

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

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

### Assessment

#### Summative

MethodPercentage contribution
Class Test  (1 hours) 10%
Coursework 20%
Written exam 70%

#### Referral

MethodPercentage contribution
Exam 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.