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, Rings, Fields and Groups
Allenby, An Introduction to Number Theory with Computing
Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers
Stark, An Introduction to Number Theory
Ireland and Rosen, A Classical Introduction to Modern Number Theory
