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2015 | Book

Why Quadratic Diophantine Equations? Read chapter Read first chapter

Quadratic Diophantine Equations

Authors: Titu Andreescu, Dorin Andrica

Publisher: Springer New York

Book Series : Developments in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory.

The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.

Table of Contents

Frontmatter
Chapter 1. Why Quadratic Diophantine Equations?
Abstract
In order to motivate the study of quadratic type equations, in this chapter we present several problems from various mathematical disciplines leading to such equations. The diversity of the arguments to follow underlines the importance of this subject.
Titu Andreescu, Dorin Andrica
Chapter 2. Continued Fractions, Diophantine Approximation, and Quadratic Rings
Abstract
The main goal of this chapter is to lay out basic concepts needed in our study in Diophantine Analysis. The first section contains fundamental results pertaining to continued fractions, some without proofs. The Theory of Continued Fractions is not new but it plays a growing role in contemporary mathematics.
Titu Andreescu, Dorin Andrica
Chapter 3. Pell’s Equation
Abstract
Euler, after a cursory reading of Wallis’s Opera Mathematica, mistakenly attributed the first serious study of nontrivial solutions to equations of the form \(x^{2} - Dy^{2} = 1\), where x ≠ 1 and y ≠ 0, to John Pell. However, there is no evidence that Pell, who taught at the University of Amsterdam, had ever considered solving such equations. They should be probably called Fermat’s equations, since it was Fermat who first investigated properties of nontrivial solutions of such equations.
Titu Andreescu, Dorin Andrica
Chapter 4. General Pell’s Equation
Abstract
This chapter gives the general theory and useful algorithms to find positive integer solutions (x, y) to general Pell’s equation (4.1.1), where D is a nonsquare positive integer, and N a nonzero integer.
Titu Andreescu, Dorin Andrica
Chapter 5. Equations Reducible to Pell’s Type Equations
Abstract
An interesting problem concerning the Pell’s equation \(u^{2} - Dv^{2} = 1\) is to study when the second component of a solution (u, v) is a perfect square.
Titu Andreescu, Dorin Andrica
Chapter 6. Diophantine Representations of Some Sequences
Abstract
In 1900, David Hilbert asked for an algorithm to decide whether a given Diophantine equation is solvable or not and put this problem tenth in his famous list of 23.
Titu Andreescu, Dorin Andrica
Chapter 7. Other Applications
Abstract
In [122] and [123] it is proven that there are infinitely many positive integers n such that 2n + 1 and 3n + 1 are both perfect squares. The proof relies on the theory of general Pell’s equations.
Titu Andreescu, Dorin Andrica
Backmatter
Metadata
Title
Quadratic Diophantine Equations
Authors
Titu Andreescu
Dorin Andrica
Copyright Year
2015
Publisher
Springer New York
Electronic ISBN
978-0-387-54109-9
Print ISBN
978-0-387-35156-8
DOI
https://doi.org/10.1007/978-0-387-54109-9

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