Ebook: Solving the Pell Equation

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.

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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.

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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.

Content:
Front Matter....Pages i-xx
Introduction....Pages 1-17
Early History of the Pell Equation....Pages 19-41
Continued Fractions....Pages 43-73
Quadratic Number Fields....Pages 75-96
Ideals and Continued Fractions....Pages 97-124
Some Special Pell Equations....Pages 125-152
The Ideal Class Group....Pages 153-184
The Analytic Class Number Formula....Pages 185-207
Some Additional Analytic Results....Pages 209-235
Some Computational Techniques....Pages 237-264
(f, p) Representations of $mathcal{O}$ -ideals....Pages 265-283
Compact Representations....Pages 285-305
The Subexponential Method....Pages 307-352
Applications to Cryptography....Pages 353-386
Unconditional Verification of the Regulator and the Class Number....Pages 387-404
Principal Ideal Testing in $mathcal{O}$ ....Pages 405-421
Conclusion....Pages 423-437
Back Matter....Pages 439-495

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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.

Content:
Front Matter....Pages i-xx
Introduction....Pages 1-17
Early History of the Pell Equation....Pages 19-41
Continued Fractions....Pages 43-73
Quadratic Number Fields....Pages 75-96
Ideals and Continued Fractions....Pages 97-124
Some Special Pell Equations....Pages 125-152
The Ideal Class Group....Pages 153-184
The Analytic Class Number Formula....Pages 185-207
Some Additional Analytic Results....Pages 209-235
Some Computational Techniques....Pages 237-264
(f, p) Representations of $mathcal{O}$ -ideals....Pages 265-283
Compact Representations....Pages 285-305
The Subexponential Method....Pages 307-352
Applications to Cryptography....Pages 353-386
Unconditional Verification of the Regulator and the Class Number....Pages 387-404
Principal Ideal Testing in $mathcal{O}$ ....Pages 405-421
Conclusion....Pages 423-437
Back Matter....Pages 439-495
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