An Introduction to Diophantine Equations A Problem-Based Approach
This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in s
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An E Introduction to Diophantine Equations A Problem-Based Approach
Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas 800 W. Campbell Road Richardson, TX 75080, USA [email protected] Ion Cucurezeanu Faculty of Mathematics and Computer Science Ovidius University of Constanta B-dul Mamaia, 124 900527 Constanta, Romania
Dorin Andrica Faculty of Mathematics and Computer Science Babeş -Bolyai University Str. Kogalniceanu 1 3400 Cluj-Napoca, Romania [email protected] and King Saud University Department of Mathematics College of Science Riyadh 11451, Saudi Arabia [email protected]
ISBN 978-0-8176-4548-9 e-ISBN 978-0-8176–4549-6 DOI 10.1007/978-0-8176-4549-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934856 Mathematics Subject Classification (2010): 11D04, 11D09, 11D25, 11D41, 11D45, 11D61, 11D68, 11-06, 97U40 © Springer Science+Business Media, LLC 2010 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
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Preface
Diophantus, the “father of algebra,” is best known for his book Arithmetica, a work on the solution of algebraic equations and the theory of numbers. However, essentially nothing is known of his life, and there has been much debate regarding precisely the years in which he lived. Diophantus did his work in the great city of Alexandria. At this time, Alexandria was the center of mathematical learning. The period from 250 bce to 350 ce in Alexandria is known as the Silver Age, also the Later Alexandrian Age. This was a time when mathematicians were discovering many ideas that led to our current conception of mathematics. The era is considered silver because it came after the Golden Age, a time of great development in the field of mathematics. This Golden Age encompasses the lifetime of Euclid.
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Preface
The quality of mathematics from this period was an inspiration for the axiomatic methods of today’s mathematics. While it is known that Diophantus lived in the Silver Age, it is hard to pinpoint the exact years in which he lived. While many references to the work of Diophantus have been made, Diophantus himself made few references to other mathematicians’ work, thus making the process of determining the time that he lived more difficult. D
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