A spectacular development in mathematics is Wiles' proof of Fermat's Last Theorem: if n>2 then xn+yn=zn has no nontrivial integer solutions. A high point of the module is a proof of Fermat's Last Theorem for n=3. Ideas relating to integer and primes are generalized to other number systems e.g. the Gaussian integers Z[i] = {x + iy | x and y integers}. An analogue of the Fundamental Theorem of Arithmetic is proved for Z[i]. Concrete numerical examples illustrate to concepts involved. 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, e.g. Fermat's Last Theorem for n=5, Mersenne primes, the abc-conjecture, recent advances.

Learning Outcomes

By the end of the module the student should be able to:

Analyze 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.

Assessment

22498-01 : Raw Module Mark : Coursework (100%)

Assessment Methods & Exceptions

2 hour Written Unseen January Examination (80%); In-course Assessment (20%).