2022/23 Undergraduate Module Catalogue

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.
- 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 website

Last updated: 10/05/2022

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