Unit name | Number Theory |
---|---|

Unit code | MATH30200 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Professor. Wooley |

Open unit status | Not open |

Pre-requisites |
MATH11511 Number Theory & Group Theory and MATH11006 Analysis 1 |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

At the end of the unit you will acquire a command of the basic tools of number theory as applicable to the investigation of congruences, arithmetic functions, Diophantine equations and beyond. In addition, you will become familiar with the underlying themes and current state of knowledge of several branches of Number Theory and its interaction with partner disciplines.

General Description of the Unit

Number theory is a thriving and active area of research whose origins are amongst the oldest in mathematics; some questions asked over two thousand years ago have not been fully answered yet! (e.g. Is there an odd perfect number?) Despite this ancient heritage, it has surprisingly contemporary applications, underpinning the internet data security that lies at the heart of the Digital Age. Although at the core of number theory one finds the basic properties of the integers and rational numbers, the subject has developed coherently in many directions as it has been influenced by (and indeed as it in turn influences) partner disciplines. Almost every conceivable mathematical discipline has played a role in this development, and indeed this web of interactions encompasses algebra and algebraic geometry, analysis, combinatorics, probability, logic, computer science, mathematical physics, and beyond

The course begins with a discussion of arithmetic functions, and with the properties and structure of congruences. The syllabus for the later part of the course changes from year to year. Amongst the applications that may be explored are Diophantine equations and elliptic curves, Diophantine approximation and transcendence, and the distribution of prime numbers. The algebraic aspects of the course are explored further in the Level 3 partner course “Algebraic Number Theory”.

Relation to Other Units

This unit develops the number theory component of the Level 4 unit Number Theory and Group Theory. The algebraic aspects of number theory are explored further in the partner Level 6 unit Algebraic Number Theory.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Learning Objectives

After completing this unit successfully, students should be able to:

- Understand and apply the basic properties of modular arithmetic so as to analyse the solubility of polynomial congruences and equations.

- Estimate average and maximal values of basic arithmetic functions.

- Exhibit some familiarity with the underlying themes and current state of knowledge of several branches of Number Theory and its interaction with partner disciplines.

Transferable Skills

- Ability to write coherent and logically sound arguments.

- Assimilation and use of novel and abstract ideas.

Lectures, homework exercises to be done by students.

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

There is no recommended text for this course. Four useful references are:

Alan Baker, A concise introduction to the theory of numbers. Cambridge University Press, 1984. xiii+95 pp. ISBN: 0-521-24383-1; 0-521-28654-9

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers. Fifth edition. John Wiley & Sons, Inc., 1991. xiv+529 pp. ISBN: 0-471-62546-9

H. E. Rose, A course in number theory. Second edition. Oxford Science Publications. The Clarendon Press, Oxford University Press, 1994. xvi+398 pp. ISBN: 0-19-853479-5; 0-19-852376-9

J. H. Silverman, A friendly introduction to number theory. Third edition. Prentice Hall, 2005, vii+434 pp. ISBN: 0-13-186137-9