A postgraduate course in Mathematics and Statistics.

Number theory has its roots in ancient history but particularly since the seventeenth century, it has undergone intensive development using ideas from many branches of mathematics. In spite of the subject’s maturity, there are still unsolved problems that are easy to state and understand – for example, is every even number greater than two the sum of two primes? In this course (and in Analytic number theory I (M823)), you’ll study number theory using techniques from analysis, in particular, the convergence of series and the calculus of residues. The course is based on readings from T.M. Apostol’s Introduction to Analytic Number Theory.

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What you will study

The Greeks were the first to classify the integers and it is to them that the first systematic study of the properties of the numbers is attributed. But after about 250 AD the subject stagnated until the seventeenth century. Since then there has been intensive development, using ideas from many branches of mathematics. There are a large number of unsolved problems in number theory that are easy to state and understand – for example:

• Is every even number greater than two the sum of two primes?
• Are there infinitely many ‘twin primes’ (primes differing by 2), such as (3, 5) or (101, 103)?
• Are there infinitely many primes of the form n² + 1?
• Does there always exist a prime between n² and (n + 1)² for every integer n > 1?

In this MSc course and in Analytic number theory I (M823), you will study number theory using techniques from analysis, in particular, the convergence of series and the calculus of residues. Among the results proved in M829 is the prime number theorem, which estimates the number of primes up to a given value x.

M829 is based on Chapters 8-14 of the set book Introduction to Analytic Number Theory by T. M. Apostol (1986, fourth edition, Springer-Verlag).

You will learn

Successful study of this course should enhance your skills in understanding complex mathematical texts, working with abstract concepts, thinking logically and constructing logical arguments, communicating mathematical ideas clearly and succinctly, and explaining mathematical ideas to others.

Entry

You must declare the MSc in Mathematics (or another qualification towards which the course can count) as your qualification intention. For this module you should already have passed Analytic number theory I (M823), whether or not you have also passed Calculus of variations and advanced calculus (M820). You should also have a sound knowledge of complex analysis, as there is a substantial amount of complex analysis in M829. An adequate preparation would be our undergraduate-level course Complex analysis (M337) (or the discontinued M332). 

All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

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

Qualifications

M829 is an optional module in our:

• Postgraduate Certificate in Mathematics (C90)
• Postgraduate Diploma in Mathematics (E23)
• MSc in Mathematics (F04)

Some postgraduate qualifications allow study to be chosen from other subject areas. 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.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

If you have a disability

The material contains small print and diagrams, which may cause problems if you find reading text difficult and you may also want to use a scientific calculator.

Study materials

What's included

Course notes, other printed materials.

You will need

We recommend that you have access to the internet at least once a week during the course and would like to point out that vital material, such as your assignments, will be delivered online.

Computing requirements

You will need a computer with internet access to study this course as it includes online activities, which you can access using a web browser.

• If you have purchased a new desktop or laptop computer since 2007 you should have no problems completing the online activities.
• If you’ve got a netbook, tablet or other mobile computing device check our Technical requirements section.
• If you use an Apple Mac you will need OS X 10.6 or later.

You can also visit the Technical requirements section for further computing information including the details of the support we provide.

Teaching and assessment

Support from your tutor

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. 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 can be found in the facts box above.

You will be expected to submit your tutor-marked assignments (TMAs) online using a special maths eTMA processor, which is used in place of the main eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper.

You will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of your answers to produce an electronic TMA would take you an unacceptably long time.

Future availability

The details given here are for the course starting in February 2013. We expect it to be next available in October 2014, and in alternate years after that, in October.

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

Student Reviews

Distance learning

The Open University is the world’s leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you’re at home, at work or on the move. As an OU student you’ll be supported throughout your studies – your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information about distance learning at the OU read Study explained.

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