Introduction

A Diophantine equation is an indeterminate equation whose unknowns are only allowed to assume integral^* or sometimes rational values. The study of such equations goes back to the ancients; indeed, they are named after Diophantus of Alexandria (c. 200–284 AD) in honour of his work on them.¹. However, it is most likely that the Greek mathematicians were investigating their properties much earlier than this. To take a simple example, consider the equation

$$x^2 + y^2 = z^2$$

((1.1))

, where we constrain a solution (x, y, z) to be a triple of integers.² Every student of high school geometry is familiar with the solution (3, 4, 5) and some are even made aware of the additional solutions (5, 12, 13) and (8, 15, 17). In fact, as we shall see below, there exists an infinitude of distinct integral solutions of (1.1) for which (x, y, z) = 1.