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Andreescu, Titu; Andrica, Dorin; Cucurezeanu, Ion

An Introduction to Diophantine Equations A Problem-Based Approach (Gebundene Ausgabe)

Springer Basel AG, Birkhäuser Boston, September 2010


345 S. - Sprache: Englisch - 20 illus. - 245x164x30 mm

ISBN: 0817645489 EAN: 9780817645489

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 presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants - including Olympiad and Putnam competitors - as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.


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Produktdetails

This book is an introduction to the study of Diophantine equations. The book, organized in two parts, introduces the reader to the main elementary methods necessary in solving Diophantine equations, presents some classical Diophantine equations, and contains complete solutions to all of the exercises. The book is intended for undergraduates, high school students and their teachers, mathematics contests participants (including Olympiad and Putnam competitors), as well as any person interested in essential mathematics. This work uniquely presents unconventional and non-routine examples, ideas, and techniques.

Inhaltsverzeichnis

Preface.-Part 1: Diophantine Equations.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat's Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell's-Type Equations.-Pell's Equation: History and Motivation.-Solving Pell's Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell's Equation.-Part 2: Solutions to Exercises and Problems.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat's Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell's-Type Equations.-Solving Pell's Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell's Equation.-References.-Index.

Kritikerstimmen

From the reviews: "Diophantus' Arithmetica is a collection of problems each followed by a solution...The book at hand is intended for high school students, undergraduates and math teachers. It is written in a language that everyone in these groups will be familiar with. The exposition is very lucid and the proofs are clear and instructive. The book will be an invaluable source for math contest participants and other math fans. It will be an excellent addition to any math library." (Alex Bogomolny, The Mathematical Association of America, October, 2010) "Diophantine analysis, the business of solving equations with integers, constitutes a subdiscipline within the larger field of number theory. One problem in this subject, Fermat's last theorem, till solved, topped most lists of the world's most celebrated unsolved mathematics problems, so the subject attracted much attention from mathematicians and even the larger public. Nevertheless, sophisticated 20th-century tools invented to attack Diophantine equations (algebraic number fields, automorphic forms, L-functions, adelic groups, etc.) have emerged as proper objects of study in their own right. So for a popular subject, modern lower-level works focused on the individual Diophantine equation (and not on big machines aimed generally at classes of such equations) are relatively rare. The present volume...fills this need...Summing Up: Recommended. Lower- and upper-division undergraduates and general readers." (D.V. Feldman, Choice, July, 2010) "This book explains methods for solving problems with Diophantine equations that often appear in mathematical competitions at various levels. ... The book can be recommended to mathematical contest participants, but also to undergraduate students, advanced high school students and teachers." (Andrej Dujella, Mathematical Reviews, Issue 2011 j)


ISBN 0-817645-48-9, ISBN 0-81-764548-9, ISBN 0-81764-548-9, ISBN 0-817-64548-9, ISBN 0-8176-4548-9

ISBN 978-0-817645-48-9, ISBN 978-0-81-764548-9, ISBN 978-0-81764-548-9, ISBN 978-0-817-64548-9, ISBN 978-0-8176-4548-9

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