Solving the Pell Equation (CMS Books in Mathematics) 2009th Edition by Michael Jacobson (PDF)

Description

Pell’s Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell’s Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation.The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell’s Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.

User’s Reviews

Editorial Reviews: Review From the reviews:”‘Solving the Pell Equation’ is a … monograph that offers encyclopedic in-depth coverage of its topic. … The book is very well-written and filled with many interesting asides. … As one of the book’s stated goals is to provide ‘a relatively gentle introduction for senior undergraduates,’ a much larger set of examples … increase the number of students at every level who could profitably read this text. … I highly recommend the book to anyone with an interest in Pell’s equation and its modern study.” (Thomas Hagedorn, The Mathematical Association of America, July, 2009)”This new book on the Pell equation, eagerly anticipated by the mathematical community and written by two active contributers to the field of computational number theory in general and to Pell’s equation in particular, exposes the ongoing interaction between modern computational number theory and practice in a way that is pleasant to read and to study, and that is readily accessible to conscientious undergraduate students. … this book is highly recommended.” (Robert Juricevic, Mathematical Reviews, Issue 2009 i) “Pell’s equation is best known for the misattribution by Euler of a method of solution to John Pell. … This work will be valuable for a comprehensive mathematics library to give strong mathematics students a motivated, deep introduction to advanced number theory. Summing Up: Recommended. Lower- and upper-division undergraduates.” (J. McCleary, Choice, Vol. 47 (5), January, 2010) From the Back Cover 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 concerning elementary number theory, but usually not in much depth. This book provides a modern and deeper approach to the problem of solving the Pell equation. The main component of this will be computational techniques, but in the process of deriving these it will be necessary to develop the corresponding theory. One objective of this book is to provide a less intimidating introduction for senior undergraduates and others with the same level of preparedness to the delights of algebraic number theory through the medium of a mathematical object that has fascinated people since the time of Archimedes. To achieve this, this work is made accessible to anyone with some knowledge of elementary number theory and abstract algebra. Many references and notes are provided for those who wish to follow up on various topics, and the authors also describe some rather surprising applications to cryptography. The intended audience is number theorists, both professional and amateur, and students, but we wish to emphasize that this is not intended to be a textbook; its focus is much too narrow for that. It could, however be used as supplementary reading for students enrolled in a second course in number theory.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Very interesting monograph on a topic in Diophantine equations.

⭐Pell’s equation refer to equations of the type x^2-d*y^2=1. These equations have for all numbers d which are not square numbers a solution. For specific values of d, already the Greek knew how to solve them. The Indian mathematician Bhaskara II described in 1150 a method which allows to solve the Pell’s equation for all nonsquare d (which is the most general case). European’s developped their methods – not being aware of Indian mathematicians – many centuries later and John Pell wrote in a translation of a book some methods to solve Pell’s equation which are now assumed to be from William Brouncker, who solved the equation for various difficult numbers d and Pell’s own work on the equation is very minor. Nevertheless, the name is kept for the equation and attempts to rename it (in order to give more credit to other mathematicians) were not successful. The book provides a detailed account of the Pell’s equation including all tools used today for solving it and also indicates why it often happens that solutions are quite big. This is a very interesting book, but for understanding most parts, some education or knowledge in mathematics is very useful.

⭐This book is the best elementary introduction to the Diophantine approximations.

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