Number Theory

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 Year 3/4 partner course “Algebraic Number Theory”, and the analytic aspects in the Year 4 partner course "Analytic Number Theory".

Relation to other units

This unit develops the number theory component of the Year 1 unit Foundations and Proof. The algebraic aspects of the course are explored further in the Year 3/4 partner course “Algebraic Number Theory”, and the analytic aspects in the Year 4 partner course "Analytic Number Theory".

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.

Syllabus

Topics covered will include:

  1. Revision of the basic properties of the integers including the Euclidean algorithm.
  2. Number-theoretic functions, especially the Möbius and Euler functions. Averages and maximum values.
  3. Congruences, including the theorems of Fermat, Euler, and Lagrange, and computational applications. The RSA cryptosystem.
  4. Primitive roots and the structure of the residues modulo m.
  5. Polynomial congruences to prime powers. Hensel’s lemma and the p-adic numbers.
  6. The quadratic residue symbols of Lagrange and Jacobi. Quadratic reciprocity.
  7. The solution of quadratic equations in integers.
  8. Introduction to one or more of the following topics, depending on time available: Diophantine approximation and transcendence, Dirichlet’s theorem on primes in arithmetic progressions, Diophantine equations and elliptic curves.

Reading and References

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 

Unit code: MATH30200
Level of study: H/6
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Professor Trevor Wooley

Pre-requisites

MATH10004 Foundations and Proof, MATH10003 Analysis 1A and MATH10006 Analysis 1B

Co-requisites

None

Methods of teaching

Lectures, homework exercises to be done by students.

Assessment methods

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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