Pell’s Equation

Pell's equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. The history of this equation is long and circuitous, and involved a number of different approaches before a definitive theory was found. There were partial patterns and quite effective methods of finding solutions, but a complete theory did not emerge until the end of the eighteenth century.
The topic is motivated and developed through sections of exercises which allow the student to recreate known theory and provide a focus for their algebraic practice. There are also several explorations that encourage the reader to embark on their own research. Some of these are numerical and often require the use of a calculator or computer. Others introduce relevant theory that can be followed up on elsewhere, or suggest problems that the reader may wish to pursue. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.
Edward Barbeau is Professor of Mathematics at the University of Toronto. He has published a number of books directed to students of mathematics and their teachers, including Polynomials (Springer 1989), Power Play (MAA 1997), Fallacies, Flaws and Flimflam (MAA 1999) and After Math (Wall & Emerson, Toronto 1995).

Pell's equation is part of a central area of algebraic number theory that treats quadratic forms and the structure of the rings of integers in algebraic number fields. It is an ideal topic to lead college students and talented high school students to a better appreciation of the power of mathematical technique. To appreciate this focused exercise book, a high school background in mathematics is all that is needed, and teachers and others interested in mathematics who do not have a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.

The Square Root of 2.- Problems Leading to Pel?s Equation and Preliminary Investigations.- Quadratic Surds.- The Fundamental Solution.- Tracking Down the Fundamental Solution.- Pel?s Equation and Pythagorean Triples.- The Cubic Analogue of Pel?s Equation.- Analogues of the Fourth and Higher Degrees.- A Finite Version of Pel?s Equation.