We use numbers in two different ways: the natural numbers 1, 2, 3... for counting; and the real numbers - fractions and decimals - for sizes, lengths, areas etc.

Number theory is about problems and methods where you just use the integers, meaning the extension of the natural numbers to include zero and negative numbers also:…-3, -2, -1, 0, 1, 2, 3…You can add, subtract and multiply integers, but you can't always divide exactly, so unless said otherwise, division means finding the quotient and remainder: m goes into n, q times with r left over, n = qm + r. If you fix m (the modulus) you can classify numbers according to the remainder r. This is called congruence or modulo arithmetic.

Addition and multiplication behave very differently in number theory. Given a number n, the equation x + y = n has lots of (integer) solutions, but xy = n has very few solutions, unless n = 0. If n is a prime number, the only solutions are x = 1, y = n or vice versa. The ancient Greeks knew, and proved, there are infinitely many prime numbers.

Number theory began with number puzzles. “Cattle problems" were ancient Babylonian and

Greek recreational maths puzzles - the Sudoku of their time. Now they are part of the number theory we meet in this module, with the name of Diophantine equations.

Number theory used to be the prize example of “useless" pure mathematics. Now, most of the security systems that protect our computers and bank accounts are built around disguising patterns by taking the remainders for some modulus m, and around the difficulty of factorising large numbers - i.e. solving xy = n with x; y > 1. E.g., even with a (simple) calculator it would take most humans quite a while to find that 4294967293 has just two prime factors. The numbers used in practical security are much bigger.

This part 1 introductory 10 credit second-year module contains sections on factorising integers, Euclid’s algorithm, congruence arithmetic, cattle problems, irrationality, and on continued fractions and quadratic equations.

• Use prime factorisation to list the divisors of a number.

• Determine highest common factors using prime factorisation and by using Euclid's Algorithm.

• Determine the integer solutions x, y of equations ax + by = c.

• Establish the irrationality of selected algebraic numbers and of quotients of logarithms.

• Apply congruences in simple arithmetical situations.

• Solve linear congruences and systems of two simultaneous linear congruences.

• Apply the simple continued fraction expansion to both rational and irrational numbers.

• Determine the periodic continued fraction expansion for square roots of integers not a perfect square.

• Solve Pell’s equation in integers.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• On-line resources that you work through at your own pace (e.g. videos, web resources, e-books, quizzes),
• On-line interactive sessions to work with other students and staff (e.g. discussions, live streaming of presentations, live-coding, group meetings)
• Face to face small group sessions (e.g. tutorials, exercise classes, feedback sessions)

Skills:
Working with congruences. Using Euclid’s algorithm. Determining whether a number is rational or not. Obtaining the first few terms of the continued fraction expansion for any real number. Solving Pell’s equation.

Transferable Skills:
Recognising patterns. Reasoning logically. Solving systems of equations modulo a positive integer m.

Formative assessment is carried out in the examples classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Orientation and Introduction.

• Divisibility and Primality.

• Highest Common Factors and Euclid's Algorithm.

• The Equation ax + by = c.

• More on Prime Numbers and Factorisation.

• Congruence Arithmetic.

• Irrationals and Fractions.

• Continued Fractions.

• Continued Fractions and Quadratic Equations.

An interactive version of this list is now available via our new reading list software: https://whelf-cardiff.alma.exlibrisgroup.com/leganto/public/44WHELF_CAR/lists/6034772580002420?auth=SAML

Jones, G.A. and Jones, J.M. 1998. Elementary number theory. Springer.

Rosen, K. H. 2011. Elementary number theory and its applications. 6th ed. McGraw Hill 

Hardy, G.H. & Wright, E.M. 2008. Introduction to the theory of numbers. 6th ed. OUP – highly recommended.

Davenport, H. 2008. The higher arithmetic. 8th ed. Cambridge – elementary.