Course Details in 2025/26 Session

Module Title	Number Theory
School	Mathematics
Department	Mathematics
Module Code	06 16214
Module Lead	P Flavell
Level	Masters Level
Credits	20
Semester	Semester 1
Pre-requisites	LC Algebra & Combinatorics 1 - (06 25659)
Co-requisites
Restrictions	None
Contact Hours	Lecture-46 hours Tutorial-10 hours Total: 56 hours
Exclusions
Description	A spectacular development in mathematics is Wiles' proof of Fermet's Last Theorem: if n>2 the x^n + y^n = z^n has no nontrivial integer solutions. A highpoint of the module is a proof of Fermat's Last Theorem for n=3. Ideas relating to integer and primes are generalised to other number systems. EG the Gaussian integers Z[I] = {x+iy| X AND Y integers}An analogue of the Fundamental Theorem of Arithmetic is proved forZ [i]. Concrete numerical examples illustrate to concepts invilved.Modular arithmetic is studied. The high point being Gauss' celebrated Law of Quadratic Reciprocity concerning the existence of square roots modulo a prime.Time permitting, other topics may be studied, eg Fermat's Last Theorem for n=5, Mersenne primes, the abc-conjecture, recent advances.
Learning Outcomes	1. That the student be able to:- analyse Diophantine equations by factorizations in appropriate rings and the use of modular arithmetic.- understand modular arithmetic and the techniques involved in proving Gauss' Law of Quadratic Reciprocity- use this law to find quadratic residues.2. By the end of the module, students should be able to explore these topics beyond the taught syllabus.
Assessment	16214-06 : Raw Module Mark : Coursework (100%)
Assessment Methods & Exceptions	3 hour Written Unseen Examination (80%); In-course Assessment (20%)
Other
Reading List	Hardy and Wright, The Theory of Numbers Allenby, An Introduction to Number Theory with Computing Ireland and Rosen, A Classical Introduction to Modern Number Theory Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers Stark, An Introduction to Number Theory Allenby, Rings, Fields and Groups