## MATH3044 Number Theory

### 15 creditsClass Size: 200

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

### Pre-requisites

MATH1021 or equivalent. Exclusion: not with MATH3171.

This module is approved as an Elective

### 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

Some of the most famous problems in mathematics involve properties of the positive integers. For example the Goldbach Conjecture is that every even integer bigger than 2 is a sum of two primes, while Fermat?s Last Theorem is that for n? 2, a sum of two nth powers cannot be an nth power. (It remained an open problem for 350 years, until proved by Wiles in 1995.) 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. The topics covered are: 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

Lectures: 26 hours. 7 examples classes.

### Methods of assessment

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

One 3 hour examination at end of semester (100%).