Pell’s Equation

1. Front Matter

   Pages i-xi

   PDF

2. The Square Root of 2

   • Edward J. Barbeau

   Pages 1-15

3. Problems Leading to Pell’s Equation and Preliminary Investigations

   • Edward J. Barbeau

   Pages 16-31

4. Quadratic Surds

   • Edward J. Barbeau

   Pages 32-42

5. The Fundamental Solution

   • Edward J. Barbeau

   Pages 43-54

6. Tracking Down the Fundamental Solution

   • Edward J. Barbeau

   Pages 55-80

7. Pell’s Equation and Pythagorean Triples

   • Edward J. Barbeau

   Pages 81-91

8. The Cubic Analogue of Pell’s Equation

   • Edward J. Barbeau

   Pages 92-112

9. Analogues of the Fourth and Higher Degrees

   • Edward J. Barbeau

   Pages 113-125

10. A Finite Version of Pell’s Equation

    • Edward J. Barbeau

    Pages 126-137

11. Back Matter

    Pages 139-213

    PDF

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, as well as some talented and motivated high school students, to a better appreciation of the power of mathematical technique. Even at the specific level of quadratic diophantine equations, there are unsolved problems, and the higher degree analogues of Pell's equation, particularly beyond the third, do not appear to have been well studied. In this focused exercise book, the topic is motivated and developed through sections of exercises which will allow the readers to recreate known theory and provide a focus for their algebraic practice. There are several explorations that encourage the reader to embark on their own research. 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 (or have forgotten) a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.

• algebra
• algebraic number field
• algebraic number theory
• diophantine equation
• number theory
• quadratic form

From the reviews:

"This book belongs to the collection ‘Problem Books in Mathematics’. Because of this choice, this book is not a course on Pell’s equation but a series of exercises which presents the theory of this equation … . This book is certainly a good book for students who are courageous enough to try to solve many exercises; they can learn a lot this way. Moreover, this long collection of exercises presents a lot of examples and a great variety of methods." (Maurice Mignotte, Zentralblatt MATH, Vol. 1030, 2004)

"This is Barbeau’s second book in the Springer Problem Books in Mathematics series … . Brief hints are given at the end of each chapter and Answers and Solutions are given at the end of the book … . All of the material in the book is aimed at undergraduate level. … The book does provide a good source of exercises and interesting ideas for student projects, as well as giving a reasonably thorough account of the solutions, applications and generalisations of Pell’s Equation." (Peter G. Brown, The Australian Mathematical Society Gazette, Vol. 30 (4), 2003)

"The present book is described by its author as a ‘focussed exercise book in algebra’, and is aimed at both college students and talented sixth-formers, with the particular intention of providing training in the key skills of manipulating algebraic expressions judiciously, and with a sense of strategy. … readers are encouraged to do much of the exploring for themselves through carefully guided examples … . The book is full of interest, and largely succeeds in its aims." (Michael Ward, The Mathematical Gazette, Vol. 88 (512), 2004)

"Ed Barbeau has prepared a ‘focused exercise book in algebra’ based on various aspects of Pell’s equation. … The author does a wonderful job in preparing and motivating the reader … . An array of interesting related problems, facts, and ‘explorations’ are found here. … This book is astepping stone towards any one of the several related areas of study, including diophantine analysis, diophantine approximation, algebraic number theory … . It is perfectly suited for any first course in number theory." (Gary Walsh, SIAM Review, Vol. 46 (1), 2004)

"The author’s book, as the title indicates, is primarily concerned with aspects of Pell’s equations … . The book has nine chapters … followed by extensive answers, comments and solutions to posed problems and explorations, a glossary of several pages, separate references to books and papers, and a brief index. … would also be suitable for any person interested in number theory. … The book reveals the enduring interest by mathematicians in Pell’s equation … ." (George W. Grossman, Mathematical Reviews, 2004f)

"Barbeau develops the theory of Pell’s equation (a piece of quadratic form theory) entirely as a series of exercises. … The book’s form recommends it for shepherding undergraduates to research, and is a good source for higher-degree analogs of Pell’s equation. Includes problem answers and solutions." (D.V. Feldman, CHOICE, December, 2003)

"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 topic is motivated and developed through sections of exercises that allow students to recreate known theory and provide a focus for their algebraic practice. There are also several explorations that encourage readers to embark on their own research." (Zentralblatt für Didaktik der Mathematik, August, 2003)

"Pell’s equations are … as old as mathematics, but the theory of these equations is a modern branch of mathematics development. … This book is a curious exercise book, but it is much more. … The exercises, explorations are well-chosen, showing the real nature of mathematical thinking … . Without any special background you willenjoy the problems throwing you in the very heart of this science. … Whoever you are … you will find some beautiful, mind awakening problems in this book." (Lajos Pintér, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

• Department of Mathematics, University of Toronto, Toronto, Canada

  Edward J. Barbeau

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