Softcover ISBN:  9781470452759 
Product Code:  AMSTEXT/48 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470456917 
Product Code:  AMSTEXT/48.E 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
Softcover ISBN:  9781470452759 
eBook: ISBN:  9781470456917 
Product Code:  AMSTEXT/48.B 
List Price:  $198.00 $148.50 
MAA Member Price:  $178.20 $133.65 
AMS Member Price:  $158.40 $118.80 
Softcover ISBN:  9781470452759 
Product Code:  AMSTEXT/48 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470456917 
Product Code:  AMSTEXT/48.E 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
Softcover ISBN:  9781470452759 
eBook ISBN:  9781470456917 
Product Code:  AMSTEXT/48.B 
List Price:  $198.00 $148.50 
MAA Member Price:  $178.20 $133.65 
AMS Member Price:  $158.40 $118.80 

Book DetailsPure and Applied Undergraduate TextsVolume: 48; 2020; 549 ppMSC: Primary 11
Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise.
The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are wellknown mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors' personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers.
Ancillaries:
ReadershipUndergraduate and graduate students interested in number theory.

Table of Contents

Cover

Title page

Copyright

Contents

Introduction

Structure of the book

Exercises

Short overview of the individual chapters

Technical details

Commemoration

Acknowledgements

Chapter 1. Basic Notions

1.1. Divisibility

Exercises 1.1

1.2. Division Algorithm

Exercises 1.2

1.3. Greatest Common Divisor

Exercises 1.3

1.4. Irreducible and Prime Numbers

Exercises 1.4

1.5. The Fundamental Theorem of Arithmetic

Exercises 1.5

1.6. Standard Form

Exercises 1.6

Chapter 2. Congruences

2.1. Elementary Properties

Exercises 2.1

2.2. Residue Systems and Residue Classes

Exercises 2.2

2.3. Euler’s Function 𝜑

Exercises 2.3

2.4. The Euler–Fermat Theorem

Exercises 2.4

2.5. Linear Congruences

Exercises 2.5

2.6. Simultaneous Systems of Congruences

Exercises 2.6

2.7. Wilson’s Theorem

Exercises 2.7

2.8. Operations with Residue Classes

Exercises 2.8

Chapter 3. Congruences of Higher Degree

3.1. Number of Solutions and Reduction

Exercises 3.1

3.2. Order

Exercises 3.2

3.3. Primitive Roots

Exercises 3.3

3.4. Discrete Logarithm (Index)

Exercises 3.4

3.5. Binomial Congruences

Exercises 3.5

3.6. Chevalley’s Theorem, Kőnig–Rados Theorem

Exercises 3.6

3.7. Congruences with Prime Power Moduli

Exercises 3.7

Chapter 4. Legendre and Jacobi Symbols

4.1. Quadratic Congruences

Exercises 4.1

4.2. Quadratic Reciprocity

Exercises 4.2

4.3. Jacobi Symbol

Exercises 4.3

Chapter 5. Prime Numbers

5.1. Classical Problems

Exercises 5.1

5.2. Fermat and Mersenne Primes

Exercises 5.2

5.3. Primes in Arithmetic Progressions

Exercises 5.3

5.4. How Big Is 𝜋(𝑥)?

Exercises 5.4

5.5. Gaps between Consecutive Primes

Exercises 5.5

5.6. The Sum of Reciprocals of Primes

Exercises 5.6

5.7. Primality Tests

Exercises 5.7

5.8. Cryptography

Exercises 5.8

Chapter 6. Arithmetic Functions

6.1. Multiplicative and Additive Functions

Exercises 6.1

6.2. Some Important Functions

Exercises 6.2

6.3. Perfect Numbers

Exercises 6.3

6.4. Behavior of 𝑑(𝑛)

Exercises 6.4

6.5. Summation and Inversion Functions

Exercises 6.5

6.6. Convolution

Exercises 6.6

6.7. Mean Value

Exercises 6.7

6.8. Characterization of Additive Functions

Exercises 6.8

Chapter 7. Diophantine Equations

7.1. Linear Diophantine Equation

Exercises 7.1

7.2. Pythagorean Triples

Exercises 7.2

7.3. Some Elementary Methods

Exercises 7.3

7.4. Gaussian Integers

Exercises 7.4

7.5. Sums of Squares

Exercises 7.5

7.6. Waring’s Problem

Exercises 7.6

7.7. Fermat’s Last Theorem

Exercises 7.7

7.8. Pell’s Equation

Exercises 7.8

7.9. Partitions

Exercises 7.9

Chapter 8. Diophantine Approximation

8.1. Approximation of Irrational Numbers

Exercises 8.1

8.2. Minkowski’s Theorem

Exercises 8.2

8.3. Continued Fractions

Exercises 8.3

8.4. Distribution of Fractional Parts

Exercises 8.4

Chapter 9. Algebraic and Transcendental Numbers

9.1. Algebraic Numbers

Exercises 9.1

9.2. Minimal Polynomial and Degree

Exercises 9.2

9.3. Operations with Algebraic Numbers

Exercises 9.3

9.4. Approximation of Algebraic Numbers

Exercises 9.4

9.5. Transcendence of 𝑒

Exercises 9.5

9.6. Algebraic Integers

Exercises 9.6

Chapter 10. Algebraic Number Fields

10.1. Field Extensions

Exercises 10.1

10.2. Simple Algebraic Extensions

Exercises 10.2

10.3. Quadratic Fields

Exercises 10.3

10.4. Norm

Exercises 10.4

10.5. Integral Basis

Exercises 10.5

Chapter 11. Ideals

11.1. Ideals and Factor Rings

Exercises 11.1

11.2. Elementary Connections to Number Theory

Exercises 11.2

11.3. Unique Factorization, Principal Ideal Domains, and Euclidean Rings

Exercises 11.3

11.4. Divisibility of Ideals

Exercises 11.4

11.5. Dedekind Rings

Exercises 11.5

11.6. Class Number

Exercises 11.6

Chapter 12. Combinatorial Number Theory

12.1. All Sums Are Distinct

Exercises 12.1

12.2. Sidon Sets

Exercises 12.2

12.3. Sumsets

Exercises 12.3

12.4. Schur’s Theorem

Exercises 12.4

12.5. Covering Congruences

Exercises 12.5

12.6. Additive Complements

Exercises 12.6

Answers and Hints

A.1. Basic Notions

A.2. Congruences

A.3. Congruences of Higher Degree

A.4. Legendre and Jacobi Symbols

A.5. Prime Numbers

A.6. Arithmetic Functions

A.7. Diophantine Equations

A.8. Diophantine Approximation

A.9. Algebraic and Transcendental Numbers

A.10. Algebraic Number Fields

A.11. Ideals

A.12. Combinatorial Number Theory

Historical Notes

Tables

Primes 2–1733

Primes 1741–3907

Prime Factorization

Mersenne Numbers

Fermat Numbers

Index

Back Cover


Additional Material

Reviews

I think the book is not only the best book on number theory, but the best textbook I have ever seen. Beginning students can gain a solid foundation in number theory, advanced students can challenge themselves with the often deep and always delightful exercises, and everyone, including experts in the field, can discover new topics or attain a better understanding of familiar ones. As with masterpieces in music in literature, one gets more out of it with each additional visit.
Béla Bajnok, Gettysburg College, The American Mathematical Monthly


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseInstructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manualExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise.
The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are wellknown mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors' personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers.
Ancillaries:
Undergraduate and graduate students interested in number theory.

Cover

Title page

Copyright

Contents

Introduction

Structure of the book

Exercises

Short overview of the individual chapters

Technical details

Commemoration

Acknowledgements

Chapter 1. Basic Notions

1.1. Divisibility

Exercises 1.1

1.2. Division Algorithm

Exercises 1.2

1.3. Greatest Common Divisor

Exercises 1.3

1.4. Irreducible and Prime Numbers

Exercises 1.4

1.5. The Fundamental Theorem of Arithmetic

Exercises 1.5

1.6. Standard Form

Exercises 1.6

Chapter 2. Congruences

2.1. Elementary Properties

Exercises 2.1

2.2. Residue Systems and Residue Classes

Exercises 2.2

2.3. Euler’s Function 𝜑

Exercises 2.3

2.4. The Euler–Fermat Theorem

Exercises 2.4

2.5. Linear Congruences

Exercises 2.5

2.6. Simultaneous Systems of Congruences

Exercises 2.6

2.7. Wilson’s Theorem

Exercises 2.7

2.8. Operations with Residue Classes

Exercises 2.8

Chapter 3. Congruences of Higher Degree

3.1. Number of Solutions and Reduction

Exercises 3.1

3.2. Order

Exercises 3.2

3.3. Primitive Roots

Exercises 3.3

3.4. Discrete Logarithm (Index)

Exercises 3.4

3.5. Binomial Congruences

Exercises 3.5

3.6. Chevalley’s Theorem, Kőnig–Rados Theorem

Exercises 3.6

3.7. Congruences with Prime Power Moduli

Exercises 3.7

Chapter 4. Legendre and Jacobi Symbols

4.1. Quadratic Congruences

Exercises 4.1

4.2. Quadratic Reciprocity

Exercises 4.2

4.3. Jacobi Symbol

Exercises 4.3

Chapter 5. Prime Numbers

5.1. Classical Problems

Exercises 5.1

5.2. Fermat and Mersenne Primes

Exercises 5.2

5.3. Primes in Arithmetic Progressions

Exercises 5.3

5.4. How Big Is 𝜋(𝑥)?

Exercises 5.4

5.5. Gaps between Consecutive Primes

Exercises 5.5

5.6. The Sum of Reciprocals of Primes

Exercises 5.6

5.7. Primality Tests

Exercises 5.7

5.8. Cryptography

Exercises 5.8

Chapter 6. Arithmetic Functions

6.1. Multiplicative and Additive Functions

Exercises 6.1

6.2. Some Important Functions

Exercises 6.2

6.3. Perfect Numbers

Exercises 6.3

6.4. Behavior of 𝑑(𝑛)

Exercises 6.4

6.5. Summation and Inversion Functions

Exercises 6.5

6.6. Convolution

Exercises 6.6

6.7. Mean Value

Exercises 6.7

6.8. Characterization of Additive Functions

Exercises 6.8

Chapter 7. Diophantine Equations

7.1. Linear Diophantine Equation

Exercises 7.1

7.2. Pythagorean Triples

Exercises 7.2

7.3. Some Elementary Methods

Exercises 7.3

7.4. Gaussian Integers

Exercises 7.4

7.5. Sums of Squares

Exercises 7.5

7.6. Waring’s Problem

Exercises 7.6

7.7. Fermat’s Last Theorem

Exercises 7.7

7.8. Pell’s Equation

Exercises 7.8

7.9. Partitions

Exercises 7.9

Chapter 8. Diophantine Approximation

8.1. Approximation of Irrational Numbers

Exercises 8.1

8.2. Minkowski’s Theorem

Exercises 8.2

8.3. Continued Fractions

Exercises 8.3

8.4. Distribution of Fractional Parts

Exercises 8.4

Chapter 9. Algebraic and Transcendental Numbers

9.1. Algebraic Numbers

Exercises 9.1

9.2. Minimal Polynomial and Degree

Exercises 9.2

9.3. Operations with Algebraic Numbers

Exercises 9.3

9.4. Approximation of Algebraic Numbers

Exercises 9.4

9.5. Transcendence of 𝑒

Exercises 9.5

9.6. Algebraic Integers

Exercises 9.6

Chapter 10. Algebraic Number Fields

10.1. Field Extensions

Exercises 10.1

10.2. Simple Algebraic Extensions

Exercises 10.2

10.3. Quadratic Fields

Exercises 10.3

10.4. Norm

Exercises 10.4

10.5. Integral Basis

Exercises 10.5

Chapter 11. Ideals

11.1. Ideals and Factor Rings

Exercises 11.1

11.2. Elementary Connections to Number Theory

Exercises 11.2

11.3. Unique Factorization, Principal Ideal Domains, and Euclidean Rings

Exercises 11.3

11.4. Divisibility of Ideals

Exercises 11.4

11.5. Dedekind Rings

Exercises 11.5

11.6. Class Number

Exercises 11.6

Chapter 12. Combinatorial Number Theory

12.1. All Sums Are Distinct

Exercises 12.1

12.2. Sidon Sets

Exercises 12.2

12.3. Sumsets

Exercises 12.3

12.4. Schur’s Theorem

Exercises 12.4

12.5. Covering Congruences

Exercises 12.5

12.6. Additive Complements

Exercises 12.6

Answers and Hints

A.1. Basic Notions

A.2. Congruences

A.3. Congruences of Higher Degree

A.4. Legendre and Jacobi Symbols

A.5. Prime Numbers

A.6. Arithmetic Functions

A.7. Diophantine Equations

A.8. Diophantine Approximation

A.9. Algebraic and Transcendental Numbers

A.10. Algebraic Number Fields

A.11. Ideals

A.12. Combinatorial Number Theory

Historical Notes

Tables

Primes 2–1733

Primes 1741–3907

Prime Factorization

Mersenne Numbers

Fermat Numbers

Index

Back Cover

I think the book is not only the best book on number theory, but the best textbook I have ever seen. Beginning students can gain a solid foundation in number theory, advanced students can challenge themselves with the often deep and always delightful exercises, and everyone, including experts in the field, can discover new topics or attain a better understanding of familiar ones. As with masterpieces in music in literature, one gets more out of it with each additional visit.
Béla Bajnok, Gettysburg College, The American Mathematical Monthly