Quadratic Diophantine Equations

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.

List of contents

Why Quadratic Diophantine Equations?.- Continued Fractions, Diophantine Approximation and Quadratic Rings.- Pell's Equation.- General Pell's Equation.- Equations Reducible to Pell's Type Equations.- Diophantine Representations of Some Sequences.- Other Applications.

Summary

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.

Additional text

“The book under review is an excellent book on the interesting subject of quadratic Diophantine equations. It is well written, well organized, and contains a wealth of material that one does not expect to find in a book of its size, with full proofs of scores of theorems. … This reviewer does not know any book that covers similar material, and sees it as a very valuable and much needed addition to the literature on number theory.” (Mowaffaq Hajja, zbMATH 1376.11001, 2018)
“Diophantine analysis aims to solve equations (usually polynomial) in integers (or rationals). … this work settles the classical foundation, then develops state-of-the-art issues, especially concerning computation. … Summing Up: Recommended. Lower-division undergraduates through professionals/practitioners.” (D. V. Feldman, Choice, Vol. 53 (9), May, 2016)
“The primary focus of this book under review is the integer solutions of Pell equations, their generalisations and related diophantine equations, along with applications of these equations. … The book is suitable for readers from the level of a motivated undergraduate upwards, who are interested in the classical techniques for solving such equations. … There is also an extensive bibliography.” (Paul M. Voutier, Mathematical Reviews, March, 2016)

Report

"The book under review is an excellent book on the interesting subject of quadratic Diophantine equations. It is well written, well organized, and contains a wealth of material that one does not expect to find in a book of its size, with full proofs of scores of theorems. ... This reviewer does not know any book that covers similar material, and sees it as a very valuable and much needed addition to the literature on number theory." (Mowaffaq Hajja, zbMATH 1376.11001, 2018)
"Diophantine analysis aims to solve equations (usually polynomial) in integers (or rationals). ... this work settles the classical foundation, then develops state-of-the-art issues, especially concerning computation. ... Summing Up: Recommended. Lower-division undergraduates through professionals/practitioners." (D. V. Feldman, Choice, Vol. 53 (9), May, 2016)
"The primary focus of this book under review is the integer solutions of Pell equations, their generalisations and related diophantine equations, along with applications of these equations. ... The book is suitable for readers from the level of a motivated undergraduate upwards, who are interested in the classical techniques for solving such equations. ... There is also an extensive bibliography." (Paul M. Voutier, Mathematical Reviews, March, 2016)

Product details

• Authors: Tit Andreescu, Titu Andreescu, Dorin Andrica
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 31.10.2016
• EAN: 9781493938803
• ISBN: 978-1-4939-3880-3
• No. of pages: 211
• Dimensions: 154 mm x 13 mm x 237 mm
• Weight: 363 g
• Illustrations: XVIII, 211 p.
• Series: Developments in Mathematics; Springer Monographs in Mathema; Developments in Mathematics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra