 Course Details in 2025/26 Session

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Module Title Number Theory Mathematics Mathematics 06 16214 P Flavell Masters Level 20 Semester 1 LC Algebra & Combinatorics 1 - (06 25659) None Lecture-46 hours Tutorial-10 hours Total: 56 hours 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. 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. 16214-06 : Raw Module Mark : Coursework (100%) 3 hour Written Unseen Examination (80%); In-course Assessment (20%) Hardy and Wright, The Theory of Numbers Allenby, An Introduction to Number Theory with Computing Ireland and Rosen, A Classical Introduction to Modern Number Theory Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers Stark, An Introduction to Number Theory Allenby, Rings, Fields and Groups