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Hardcover ISBN:  9781470430979 
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Hardcover ISBN:  9781470430979 
Product Code:  AMSTEXT/31 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470446727 
Product Code:  AMSTEXT/31.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470430979 
eBook ISBN:  9781470446727 
Product Code:  AMSTEXT/31.B 
List Price:  $174.00$131.50 
MAA Member Price:  $156.60$118.35 
AMS Member Price:  $139.20$105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 31; 2018; 314 ppMSC: Primary 11; 37;
This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) openended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.
The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.ReadershipUndergraduate students interested in number theory.

Table of Contents

Cover

Title page

Contents

Preface

Note to the Instructor

Organization

Acknowledgments

Introduction

Chapter 1. Integers

1. The Integers and the Well Ordering Property

2. Divisors and the Division Algorithm

3. Greatest Common Divisor and the Euclidean Algorithm

4. Prime Numbers and Unique Factorization

Exercises

Chapter 2. Modular Arithmetic

1. Basic Arithmetic

2. Inverses and Fermat’s Little Theorem

3. Linear Congruences and the Chinese Remainder Theorem

Exercises

Chapter 3. Quadratic Reciprocity and Primitive Roots

1. Quadratic Reciprocity

2. Computing 𝑚th Roots Modulo 𝑛

3. Existence of Primitive Roots

Exercises

Chapter 4. Secrets

1. Basic Ciphers

2. Symmetric Ciphers

3. Diffie–Hellman Key Exchange

4. Public Key Cryptography (RSA)

5. Hash Functions and Check Digits

6. Secret Sharing

Exercises

Chapter 5. Arithmetic Functions

1. Euler Totient Function

2. Möbius Function

3. Functions on Divisors

4. Partitions

Exercises

Chapter 6. Algebraic Numbers

1. Algebraic or Transcendental

2. Quadratic Number Fields and Norms

3. Integers, Divisibility, Primes, and Irreducibles

4. Application: Sums of Two Squares

Exercises

Chapter 7. Rational and Irrational Numbers

1. Diophantine Approximation

2. Height of a Rational Number

3. Heights and Approximations

4. Continued Fractions

5. Approximating Irrational Numbers with Convergents

Exercises

Chapter 8. Diophantine Equations

1. Introduction and Examples

2. Working Modulo Primes

3. Pythagorean Triples

4. Fermat’s Last Theorem

5. Pell’s Equation and Fundamental Units

6. Waring Problem

Exercises

Chapter 9. Elliptic Curves

1. Introduction

2. Addition of Points

3. Points of Finite Order

4. Integer Points and the Nagel–Lutz Theorem

5. Mordell–Weil Group and Points of Infinite Order

6. Application: Congruent Numbers

Exercises

Chapter 10. Dynamical Systems

1. Discrete Dynamical Systems

2. Dynatomic Polynomials

3. Resultant and Reduction Modulo Primes

4. Periods Modulo Primes

5. Algorithms for Rational Periodic and Preperiodic Points

Exercises

Chapter 11. Polynomials

1. Introduction to Polynomials

2. Factorization and the Euclidean Algorithm

3. Modular Arithmetic for Polynomials

4. Diophantine Equations for Polynomials

Exercises

Bibliography

List of Algorithms

List of Notation

Index

Back Cover


Additional Material

Reviews

If you see the value of stressing calculation and computers in a first course in number theory, then this book is one that you will want to take a good look at the next time you teach number theory.
Mark Hunacek, MAA Reviews


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 Book Details
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This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) openended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.
The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.
Undergraduate students interested in number theory.

Cover

Title page

Contents

Preface

Note to the Instructor

Organization

Acknowledgments

Introduction

Chapter 1. Integers

1. The Integers and the Well Ordering Property

2. Divisors and the Division Algorithm

3. Greatest Common Divisor and the Euclidean Algorithm

4. Prime Numbers and Unique Factorization

Exercises

Chapter 2. Modular Arithmetic

1. Basic Arithmetic

2. Inverses and Fermat’s Little Theorem

3. Linear Congruences and the Chinese Remainder Theorem

Exercises

Chapter 3. Quadratic Reciprocity and Primitive Roots

1. Quadratic Reciprocity

2. Computing 𝑚th Roots Modulo 𝑛

3. Existence of Primitive Roots

Exercises

Chapter 4. Secrets

1. Basic Ciphers

2. Symmetric Ciphers

3. Diffie–Hellman Key Exchange

4. Public Key Cryptography (RSA)

5. Hash Functions and Check Digits

6. Secret Sharing

Exercises

Chapter 5. Arithmetic Functions

1. Euler Totient Function

2. Möbius Function

3. Functions on Divisors

4. Partitions

Exercises

Chapter 6. Algebraic Numbers

1. Algebraic or Transcendental

2. Quadratic Number Fields and Norms

3. Integers, Divisibility, Primes, and Irreducibles

4. Application: Sums of Two Squares

Exercises

Chapter 7. Rational and Irrational Numbers

1. Diophantine Approximation

2. Height of a Rational Number

3. Heights and Approximations

4. Continued Fractions

5. Approximating Irrational Numbers with Convergents

Exercises

Chapter 8. Diophantine Equations

1. Introduction and Examples

2. Working Modulo Primes

3. Pythagorean Triples

4. Fermat’s Last Theorem

5. Pell’s Equation and Fundamental Units

6. Waring Problem

Exercises

Chapter 9. Elliptic Curves

1. Introduction

2. Addition of Points

3. Points of Finite Order

4. Integer Points and the Nagel–Lutz Theorem

5. Mordell–Weil Group and Points of Infinite Order

6. Application: Congruent Numbers

Exercises

Chapter 10. Dynamical Systems

1. Discrete Dynamical Systems

2. Dynatomic Polynomials

3. Resultant and Reduction Modulo Primes

4. Periods Modulo Primes

5. Algorithms for Rational Periodic and Preperiodic Points

Exercises

Chapter 11. Polynomials

1. Introduction to Polynomials

2. Factorization and the Euclidean Algorithm

3. Modular Arithmetic for Polynomials

4. Diophantine Equations for Polynomials

Exercises

Bibliography

List of Algorithms

List of Notation

Index

Back Cover

If you see the value of stressing calculation and computers in a first course in number theory, then this book is one that you will want to take a good look at the next time you teach number theory.
Mark Hunacek, MAA Reviews