Code 22498
Level of study Third/Final year
Credit value 20
Semester Students may study either 1 or both.
Pre-requisite modules 22481
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.