Solving the Pell Equation
Pell's equation is a very simple, yet fundamental Diophantine equation which is believed to have been known to mathematicians for over 2000 years. Because of its popularity, the Pell equation is often discussed in textbooks and recreational books concerni
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Michael J. Jacobson, Jr. Hugh C. Williams
Solving the Pell Equation
Canadian Mathematical Society Société mathématique du Canada
Canadian Mathematical Society Société mathématique du Canada Editor s-in-Chief Rédacteurs-en-chef K. Dilcher K. Taylor Advisory Board Comité consultatif P. Borwein R. Kane S. Shen
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Michael J. Jacobson, Jr. Hugh C. Williams
Solving the Pell Equation
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Michael J. Jacobson, Jr. Department of Computer Science University of Calgary 2500 University rive N Calgary AB T2N 1N4 Canada [email protected]
Hugh C. illiams Department of Mathematics and Statistics University of Calgary 2500 University rive N Calgary AB T2N 1N4 Canada [email protected]
Editors-in-Chief Rédacteurs-en-chef Karl Dilcher K. Taylor Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia B3H 3J5 Canada [email protected]
ISBN 978-0-387-84922-5 e-ISBN 978-0-387-84923-2 I 10.1007 978-0-387-84923-2 Library of Congress Control Number: 2008939034 Mathematics Subject Classification (2000): 11D09 11A55 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
To the children: Amanda, Alexa, Hannah, Graeme, and Sarah.
Preface
Let D be a positive non-square integer. The misnamed Pell equation is an expression of the form (0.1) T 2 − DU 2 = 1 , where T and U are constrained to be integers. For example, if D = 13, then T = 649 and U = 180 is a solution of (0.1). This very simple Diophantine equation seems to have been known to mathematicians for over 2000 years. Indeed, there is very strong evidence that it was known to Archimedes, as the Cattle Problem, attributed to him in antiquity, makes very clear. Even today, research involving this equation continues to be very active; at least 150 articles dealing with it in various contexts have appeared within the last decade. One of the main reasons for this interest is that the equation has a habit of popping up in a variety of surprising settings; it is also of great importance in solving the general second-degree Diophantine equation in two unknowns: ax2 + bxy + cy 2 + dx + ey + f = 0 . Furthermore, the problem of solving (0.1) is connected to that of determining the regulator, an important i
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