| Module Title | Number Theory |
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| School | Mathematics |
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| Department | Mathematics |
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| Module Code | 06 22498 |
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| Module Lead | P Flavell |
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| Level | Honours Level |
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| Credits | 20 |
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| Semester | Full Term |
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| Pre-requisites |
LC Algebra & Combinatorics 1 - (06 25659)
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| Co-requisites |
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| Restrictions | None |
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| Contact Hours |
Lecture-46 hours
Tutorial-10 hours
Guided independent study-144 hours
Total: 200 hours
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| Exclusions | |
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| 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. |
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| 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.
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| Assessment |
22498-01 : Raw Module Mark : Coursework (100%)
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| Assessment Methods & Exceptions | 3 hour Written Unseen Examination (80%); In-course Assessment (20%). |
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| Other | None |
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| Reading List |
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