*Number theory* looks at some classical problems concerning the integers, including the solution of Diophantine equations; the distribution of prime numbers; the theory of congruences; quadratic reciprocity; and the theory of continued fractions. *Mathematical logic* sets out to prove Gödel’s incompleteness theorem, a result of philosophical importance for the limits of mathematical proof. To lay the ground for this theorem we look first at apparently different notions of computability that all in fact coincide, and then discuss a formal proof system for basic number theory. To study this module you should have a sound knowledge of relevant mathematics provided by the appropriate Level 2 study.

The module will give you an insight into two branches of very pure mathematics that have both historical and philosophical significance. By the end of it you should feel confident to tackle number-theoretic problems and have an appreciation of the nature and limitations of mathematics.

The module consists of two independent sections that are studied concurrently. Each has its own course texts and written work.

*Number theory* ** **This section is concerned with the integers, and in particular with the solution of classical problems that require integer solutions. It begins by considering some elementary properties of the integers, such as divisibility and greatest common divisors. This leads to a method of solving the linear Diophantine equation ax + by = c, that is, finding solutions to the equation that are integers.

Every integer greater than 1 is shown to be a unique product of primes, and some results are obtained concerning the distribution of primes among the integers. In the theory of congruences, methods are developed for solving linear congruences such as ax ≡ b {mod n) and the classical theorems of Fermat and Wilson are obtained. We then consider multiplicative functions: functions f satisfying f(m) x f(n) = f(mn) for relatively prime integers m and n, and in particular Euler’s φ-function, which counts the number of integers in the set { 0, 1, …, (n–1)} that are relatively prime to n. Returning to congruences we consider the solution of quadratic congruences, which leads to Gauss’s law of quadratic reciprocity. Finally, the story of continued fractions is developed and applied as a method of solving further examples of Diophantine equations.

*Mathematical logic *This section looks at theoretical issues concerning algorithms and what they can compute, and at a theory of mathematical reasoning. One of the major questions we address is whether there is an algorithm for deciding which statements of number theory are true. On the way to answering this, we discuss two different abstract notions of computable functions: those arising from unlimited register machines and those from the theory of recursive functions. First we show that these notions of computability give rise to the same class of computable functions, give evidence towards Church’s thesis on computability and lead to results about limitations on what can be computed. Then we look at the formalisation of a mathematical language for number theory, and at a formal proof system for it. Finally, the material so far is combined to give proofs of Gödel’s incompleteness theorem, a result of great philosophical importance for the limits of mathematical endeavour.

Successful study of this module should enhance your skills in understanding complex mathematical texts, thinking logically and constructing logical arguments.

This is a Level 3 module. Level 3 modules build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with The Open University.

Although the module assumes no specific mathematical knowledge beyond A-level or Scottish Highers in pure mathematics, mastering the concepts requires considerable mathematical sophistication, such as facility in reading mathematical arguments and experience of producing them as developed in either of our level 2 mathematics modules Pure mathematics (M208) or Mathematical methods and models (MST209)*.* Studying either or preferably both of these Level 2 modules will give you the necessary background and skills for this Level 3 module.

If you have any doubt about the suitability of the module, please contact our Student Registration & Enquiry Service.

There is no specific preparatory work required for this module. A flavour of what is involved in number theory can be obtained from books such as *Elementary Number Theory *by D.Burton,* *or *Elementary Number Theory *by* *Gareth Jones and Mary Jones*. *A flavour of the first part of the mathematical logic can be obtained from the book *Computability *by N. Cutland.

M381 is an optional module in our:

- #BSc (Hons) Computing and Mathematical Sciences [B14]#
- BSc (Hons) Mathematics (B31)
- BA (Hons) or BSc (Hons) Mathematics and Statistics (B36)
- BSc (Hons) Mathematics and its Learning (B46)
- BSc (Hons) Computing & IT and a second subject (B67) - where the second subject is Mathematics

It can also count towards most of our other degrees at bachelors level, where it is equally appropriate to a BA or BSc. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

Sometimes you will not be able to count a module towards a qualification if you have already taken another module with similar content. To check any excluded combinations relating to this module, visit our excluded combination finder or check with our Student Registration & Enquiry Service before registering.

Written transcripts are available for the audio-visual material. This module may be substantially challenging if you have impaired sight, but it is not impossible. Our Services for disabled students website has the latest information about availability.

Module books.

A calculator would be useful, though it is not essential. A simple four-function (+ – x ÷) model would suffice.

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You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.

Contact our Student Registration & Enquiry Service if you want to know more about study with the Open University before you register.

The assessment details for this module can be found in the facts box above.

Please note that tutor-marked assignments (TMAs) for all undergraduate mathematics and statistics modules must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

The details given here are for the module that starts in October 2013 when it will be available for the last time. A 60-credit replacement module, Further pure mathematics (M303), is available from October 2014.