Module Title  Number Theory 
School  Mathematics 
Department  Mathematics 
Module Code  06 22498 
Module Lead  P Flavell 
Level  Honours Level 
Credits  20 
Semester  Full Term 
Prerequisites 
LC Algebra & Combinatorics 1  (06 25659)
LI Algebra & Combinatorics 1  (06 27363)

Corequisites 

Restrictions  None 
Contact Hours 
Lecture46 hours
Tutorial10 hours
Guided independent study144 hours
Total: 200 hours

Exclusions  
Description  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 abcconjecture, 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 
2249801 : Raw Module Mark : Coursework (100%)

Assessment Methods & Exceptions  Assessments: 2249801: Exam : Exam (CT)  Written Unseen (90%)
2249802 : Continuous Assessment : Coursework or class Tests (10%)

Other  None 
Reading List 
