Solving the Pell Equation pp 1-17 | Cite as

# Introduction

Chapter

A Diophantine equation is an indeterminate equation whose unknowns are only allowed to assume integral, where we constrain a solution (

^{*}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)

*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.## Keywords

Fundamental Solution Trivial Solution Diophantine Equation Triangular Number Greek Mathematician
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer Science+Business Media, LLC 2009