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Module Title Number Theory
SchoolMathematics
Department Mathematics
Module Code 06 22498
Module Lead P Flavell
Level Honours Level
Credits 20
Semester Full Term
Pre-requisites LC Algebra & Combinatorics 1 - (06 25659) LI Algebra & Combinatorics 1 - (06 27363)
Co-requisites
Restrictions None
Contact Hours Lecture-46 hours
Tutorial-10 hours
Guided independent study-144 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 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 Assessments: 22498-01: Exam : Exam (CT) - Written Unseen (90%)
22498-02 : Continuous Assessment : Coursework or class Tests (10%)
Other None
Reading List