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 module (and in Analytic number theory II (M829)), you’ll study number theory using techniques from analysis, in particular, the convergence of series and the calculus of residues. The module is based on readings from T.M. Apostol’s Introduction to Analytic Number Theory.
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01 Oct 2022 |
Jun 2023 |
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Registration closes 08/09/22 (places subject to availability) Click to register
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This module is expected to start for the last time in October 2027.
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 AD 250 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 n 2 + 1?
- Does there always exist a prime between n 2 and (n + 1)2 for every integer n > 1?
In this MSc module (and in Analytic number theory II (M829)), 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 this module are:
- Dirichlet’s theorem on primes in an arithmetic progression, which states that there are infinitely many primes in a progression such as 1, 5, 9, 13, 17 …
- the law of quadratic reciprocity, which compares the solubility of the congruences x2 ≡ p(mod q) and x2 ≡ q(mod p), where p and q are primes.
This module is based on selected readings from the set book Introduction to Analytic Number Theory by T. M. Apostol. It covers most of the material in the first seven chapters, and part of Chapter 9.
You will learn
Successful study of this module 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.
This module and Calculus of variations and advanced calculus (M820) are entry-level modules for the MSc in Mathematics (F04), and normally you should have studied one of them before progressing to the intermediate and advanced intermediate modules.
Note you must have completed this module before studying Analytic number theory II (M829).
Entry
You must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention.
You should normally have a minimum of a 2:2 honours degree in mathematics or a 2:1 honours degree in a subject with a high mathematical content. If you don’t have such a qualification, your application will be considered and you may be asked to complete an entry test. Non-graduates will not normally be admitted to M823 unless as part of another Open University qualification. Students already registered for a qualification of which M823 is a constituent part will normally be admitted to M823.
You should have a good background in pure mathematics, with some experience in number theory and analysis. An adequate preparation would be our undergraduate-level modules Pure mathematics (M208) and Further pure mathematics (M303). A knowledge of complex analysis (as in, for example, Complex analysis (M337)) would be an advantage, but is not necessary. Note that if you wish later to study Analytic number theory II (M829), then knowledge of complex analysis is a requirement.
Whatever your background, you should assess your suitability for this MSc in Mathematics module by trying our diagnostic quiz.
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 module, please speak to an adviser.
Qualifications
M823 is an optional module in our:
This module can also count towards F12 and M03, which are no longer available to new students.
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.
To find out more about what kind of support and adjustments might be available, contact us or visit our Disability support website.
Study materials
What's included
You’ll be provided with printed course notes covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques that are contained in the set book.
You’ll also have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- assessment details and submission section
- online tutorial access
- access to student and tutor group forums.
You will need to obtain your own copy of the set book. Only the set book as printed by the publisher will be permitted in the examination, and not a version you have printed yourself.
Computing requirements
You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11), or macOS (10.15 or higher).
Any additional software will be provided or is generally freely available.
To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).
Our module websites comply with web standards, and any modern browser is suitable for most activities.
Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.
It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop as described above.
Materials to buy
Set books
- Apostol, T.M. Introduction to Analytic Number Theory Springer £46.99 - ISBN 9780387901633 This book is Print on Demand and can be ordered through any bookseller. Please allow at least 2 weeks for receipt following order.
Teaching and assessment
Support from your tutor
Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:
- Marking your assignments (TMAs) and providing detailed feedback for you to improve.
- Guiding you to additional learning resources.
- Providing individual guidance, whether that’s for general study skills or specific module content.
The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.
Assessment
The assessment details can be found in the facts box.
Students also studied
Students who studied this module also studied at some time:
Future availability
Analytic number theory I (M823) starts once a year – in October.
This page describes the module that will start in October 2022.
We expect it to start for the last time in October 2027.
How to register
To register a place on this module return to the top of the page and use the Click to register button.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Student Policies and Regulations website.