Number theory and mathematical logic
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| Examination | No residential school |
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Summary
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.
Course content
The course 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 course 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.
Entry
This is a Level 3 course. Level 3 courses 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. The course assumes no specific mathematical knowledge beyond A-level or Scottish Highers in pure mathematics, but many of the concepts require considerable mathematical sophistication, such as facility in reading mathematical arguments and some experience of producing them, as developed in either of the Open University Level 2 mathematics courses Pure mathematics (M208) or Mathematical methods and models (MST209). Students are more likely to complete this course successfully if they have acquired their prerequisite knowledge through passing these courses. If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.
Preparatory work
There is no specific preparatory work required for this course. 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.
Qualifications
M381 is an optional course in our
- BA/BSc (Honours) Computing and Mathematical Sciences (B14)
- BA/BSc (Honours) Mathematics (B31)
- BA/BSc (Honours) Mathematics and Statistics (B36)
- BSc (Honours) Mathematics and its Learning (B46)
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 course can count towards these qualifications because from time to time the structure and requirements may change.
Excluded combinations
Sometimes you will not be able to count a course towards a qualification if you have already taken another course with similar content. To check any excluded combinations relating to this course, visit our excluded combination finder or check with our Student Registration & Enquiry Service before registering.
If you have a disability or additional requirement
Written transcripts are available for the audio-visual material. This course 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.
If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our booklet Meeting Your Needs which you can download or request from our Student Registration & Enquiry Service.
You can also find information about accessible course materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.
Course materials
What's included
Course books.
You will need
A calculator would be useful, though it is not essential. A simple four-function (+ – x ÷) model would suffice.
Computing requirements
This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.
You will need internet access and a computer. If you have purchased a new computer since 2002 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.
Teaching and assessment
Support from your tutor
You will have a tutor who will help you with the course 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 course. Contact our Student Registration & Enquiry Service if you want to know more about study with the Open University before you register.
Assessment
The assessment details for this course can be found in the facts box above.
Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.
Assessment is an essential part of the teaching, so you are expected to complete it all. But if you unavoidably miss or do badly in an assignment some courses allow you a ‘substitution score’. In M381 this rule can apply to one assignment only. You will be given more detailed information when you begin the course.
Professional recognition
This course may help you to gain recognition from a professional body. You can download Recognition leaflet 3.6 Institute of Mathematics and its Applications or ask our Student Registration & Enquiry Service for a copy.
Students also studied
Students who studied this course also studied at some time:
Future availability
The details given here are for the course that starts in February 2009. Thereafter it will be available at the same time in alternate years – the course will be offered again in 2011.
How to register
To register a place on this course return to the top of the page and use the Click to register button. For more information and advice about registration see OU Study Explained.
About this page
An undergraduate course in Mathematics and Statistics.
Study explained
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Student Reviews
Of all the maths courses I have studied with the OU so far (and I have done a fair few), ...
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This course was a mixed bag for me, so firstly I will state that the course material was excellent (which ...
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We're delighted that this student has enjoyed the course! Although the course, like almost all other maths courses, has no ...
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