## MATH3044 Number Theory

### 15 creditsClass Size: 200

Module manager: Professor J.K. Truss
Email: j.k.truss@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisites

 MATH1022 Introductory Group Theory

### This module is mutually exclusive with

 MATH3171 Algebra and Numbers

This module is approved as an Elective

### 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 7 1.00 7.00 Lecture 26 1.00 26.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

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