Introduction to number theory

Introduction

Please note: a Statement of Participation is not issued for this course.

This free OpenLearn course, Introduction to number theory, is an extract from the Open University module MST125 Essential mathematics 2 [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] . The module builds on mathematical ideas introduced in MST124 Essential mathematics 1, covering a wide range of topics from different areas of mathematics. The practical application to problems provides a firm foundation for further studies in mathematics and other mathematically rich subjects such as physics and engineering. Topics covered include mathematical typesetting, number theory, conics, statics, geometric transformations, calculus, differential equations, mathematical language and proof, dynamics, eigenvalues and eigenvectors and combinatorics. It also helps develop the abilities to study mathematics independently, to solve mathematical problems and to communicate mathematics.

Introduction to number theory consists of material from MST125 Unit 3, Number theory and has three study sections in total. You should set aside approximately 6 hours to study each of the sections; the whole extract should take about 18 hours to study. The extract is a small part (around 8%) of a large module that is studied over eight months, and so can give only an approximate indication of the level and content of the full module.

The extract is relatively self-contained and should be reasonably easy to understand for someone who has not previously studied any of the texts in MST125. These include a review of key techniques and an introduction to mathematical typesetting, therefore a fluency with the rules of arithmetic, highest common factors (HCF), basic algebra including subscript notation and inequalities is essential for this extract.

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 MST125 software or to video material (although please note that the PDF may contain references to other parts of MST125). In this extract, some illustrations have also been removed due to copyright restrictions.

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Number theory is a branch of mathematics concerned with the properties of integers, which can be traced back at least to the Ancient Greeks. There are many famous unsolved problems, including Goldbach’s conjecture, which keep mathematicians busy. Another famous conjecture that you may well have heard of, Fermat’s last theorem, wasn’t proved until fairly recently, in 1994.

Section 1 introduces you to Euclid’s algorithm, which is used to find the HCF of two integers, and introduces the idea of congruences, which are mathematical statements used to compare remainders when two integers are each divided by another integer.

Section 2 introduces modular, or clock, arithmetic in which the usual arithmetic operations of addition, subtraction, multiplication and division are applied to congruences. The section concludes by looking at divisibility tests and digit checking which is used to help prevent errors in ID numbers.

Section 3 introduces multiplicative inverses, which provide a method for division in modular arithmetic, and their use in solving linear congruences which are used in cryptography for disguising information or ciphers. The section concludes by looking at how to unravel particular types of ciphers, called affine ciphers, by solving linear congruences.