## MATH3044 Number Theory

### 15 creditsClass Size: 120

Module manager: Dr Alison Parker
Email: a.e.parker@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### 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.
- 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