Number theory

Introduction

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This free OpenLearn course, Number theory, is an extract from the Open University course M303 Further pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level course that introduces important topics in the theory of pure mathematics including: number theory; the algebraic theory of rings and fields; and metric spaces. Students studying M303 develop their understanding of group theory and real analysis and see how some of the ideas are applied to cryptography and fractals.

Number theory consists of material from M303 Book A, Chapters 1 and 2 and has four sections in total. You should set aside about four hours to study each of the sections; the main content though is in sections 2, 3 and 4, so you may want to spend a bit longer on these three sections. The whole extract should take about 16 hours to study. The extract is a small part (around 4%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

The aim of this extract is to lay a firm foundation for the study of number theory, which is all about problem-solving. You cannot get to grips with this subject without solving lots of problems yourself. The extract includes lots of problems and while studying you should always have pencil and paper at hand. It is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of pure mathematics, such as could be gained from studying the Open University course M208 Pure mathematics. A few techniques and definitions are present in the extract without explanation.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M303 video material (although please note that the PDF may contain references to other parts of M303). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 provides a brief introduction to the kinds of problem that arise in Number Theory.

Section 2 reviews a powerful method of proof in number theory: proof by mathematical induction. You may well have met this idea before; a more formal approach is taken here.

Section 3 introduces and makes precise the key notion of divisibility. For this we use a result, the Division Algorithm, concerning the division of one integer by another. Its consequences, both practical and theoretical, make it a cornerstone of number theory.

Section 4 explores some of the basic properties of the prime numbers and introduces the sieve of Eratosthenes, a relatively simple way of listing all the primes up to some number.