School of Mathematics and Statistics Home | About the school | Contact | Courses | Research | Personnel list

Courses in Mathematics and Statistics Level 1 Modules Level 2 Modules Level 3 Modules Level 4 Modules Level 5 Modules Honours timetable 2009/2010 Sem. 1 2009/2010 Sem. 2 2010/2011 Sem. 1 2010/2011 Sem. 2

MT4519 NUMBER THEORY Aims This course examines some interesting properties of the ring Z of integers. The subject has been studied since the earliest times (the Babylonians and Ancient Greeks made notable contributions) and has exercised the talents of some of the greatest mathematicians (Diophantus, Fermat, Euler, Gauss, Hilbert, ... ). With the widespread use of computers much of what a few years ago could be regarded as a dry academic study has come back into the mainstream of mathematics. As G H Hardy and E M Wright wrote in the Preface to An Introduction to the Theory of Numbers (1938): ... the subject matter is so attractive that only extravagant incompetence could make it dull. Syllabus Prime numbers Basic properties, Applications of group theory, Fermat and Mersenne primes, Distribution of primes, Factorisation, Cryptography Other algebraic systems Gaussian integers, Related systems, Lagrange's four square theorem Modular arithmetic Some group theory, Primitive roots, Quadratic residues, Quadratic reciprocity, Square roots modulo non-primes Continued fractions Introduction, Approximating real numbers by rationals, Continued fractions of square roots, Pell's equation Pythagorean triples Quadratic forms Textbooks R B J T Allenby and E J Redfern, Introduction to number theory with computing (Edward Arnold 1989) H Davenport, The Higher Arithmetic, an introduction to Number Theory (CUP 1st ed. 1952, 8th ed. 2006) A Baker, A concise introduction to the theory of numbers (CUP 1990) A booklet of notes will be given out near the start of the course. Assessment 2 Hour Examination = 100% Prerequisites One of MT3501, MT3503 or MT3504 Availability Academic year 2009/10 in semester 2 at 10 Click for the Honours timetables for: 2009/2010 Semester 1 2009/2010 Semester 2 2010/2011 Semester 1 2010/2011 Semester 2 Lecturer Dr J J O'Connor Click here for access to past examination papers for this module. Click here to see the University Course Catalogue entry. Revised: JOC (January 2010)