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