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Module Title Number Theory Mathematics Mathematics 06 22498 P Flavell Honours Level 20 Semester 1 LC Algebra & Combinatorics 1 - (06 25659) None Guided independent study-144 hours Tutorial-10 hours Lecture-46 hours Total: 200 hours 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. 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. 22498-01 : Raw Module Mark : Coursework (100%) Assessment: Online January assessment (50%); In-course assessment (50%). None