Unit information: Number Theory in 2020/21

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	Dr. Najnudel
Open unit status	Not open
Pre-requisites	MATH10004 Foundations and Proof (or MATH10010 Introduction to Proofs and Group Theory), MATH10003 Analysis 1A and MATH10006 Analysis 1B (or MATH10011 Analysis)
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

By 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.

Unit Description

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 unit Introduction to Proofs and Group Theory. The algebraic aspects of number theory are explored further in the partner Level M/7 unit Algebraic Number Theory.

Intended Learning Outcomes

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.

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

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.

Reading and References

Recommended

• Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge' University'Press, 1984. pp. xiii+95
• Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory of Numbers, Fifth edition, John Wiley & Sons, Inc., 1991. pp. xiv+529
• H. E. Rose, A Course in Number Theory. Second edition. Oxford Science Publications. The Clarendon Press, Oxford University Press, 1994. pp. xvi+398
• Joseph H. Silverman, A Friendly Introduction to Number Theory, Third edition. Prentice Hall, 2005, pp. vii+434