# 2023/24 Undergraduate Module Catalogue

## MATH3044 Number Theory

### 15 creditsClass Size: 120

**Module manager:** Dr Vincenzo L Mantova**Email:** V.L.Mantova@leeds.ac.uk

**Taught:** Semester 2 (Jan to Jun) View Timetable

**Year running** 2023/24

### Pre-requisite qualifications

MATH2020 or MATH2022 or equivalent**This module is approved as a discovery module**

### Module summary

This module is mainly about the work of the 18th Century mathematicians Euler, Lagrange and Gauss, including such highlights as Lagrange's Theorem that every positive integer is a sum of at most four squares, and Gauss's Law of quadratic reciprocity. We shall also introduce continued fractions to help solve Pell's equation.### Objectives

To introduce some of the main results and methods of elementary number theory.On completion of this module, students should be able to:

a) work with divisors, primes and prime factorizations, and use the Euclidean algorithm;

b) compute with congruences, including using Fermat's and Euler's theorems;

c) use primitive roots and other methods to test numbers for primality;

d) calculate Legendre symbols using quadratic reciprocity and other methods;

e) use continued fractions to solve Pell's equation and to approximate reals by rationals.

### Syllabus

- Prime factorization and applications.

- Congruences.

- Fermat's Little Theorem and its use in looking for prime factors.

- Euler's function. Wilson's Theorem.

- Pythagorean triples.

- Integers which are sums of 2,3,4 squares.

- Fermat's conjecture for Primitive roots.

- Quadratic reciprocity and applications.

- Gaussian integers and various generalisations.

- Use in solving certain Diophantine equations.

- Continued fractions.

- 'Best' approximation of reals by rationals. Pell's equation.

- Brief explanation of the principles behind public key cryptography.

### Teaching methods

Delivery type | Number | Length hours | Student hours |

Lecture | 33 | 1.00 | 33.00 |

Private study hours | 117.00 | ||

Total Contact hours | 33.00 | ||

Total hours (100hr per 10 credits) | 150.00 |

### Private study

Studying and revising of course material.Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments### Methods of assessment

**Exams**

Exam type | Exam duration | % of formal assessment |

Standard exam (closed essays, MCQs etc) | 2 hr 30 mins | 100.00 |

Total percentage (Assessment Exams) | 100.00 |

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

### Reading list

The reading list is available from the Library websiteLast updated: 15/01/2024

## Browse Other Catalogues

- Undergraduate module catalogue
- Taught Postgraduate module catalogue
- Undergraduate programme catalogue
- Taught Postgraduate programme catalogue

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