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Number Theory
 
Róbert Freud University Eötvös Loránd, Budapest, Hungary
Number Theory
Softcover ISBN:  978-1-4704-5275-9
Product Code:  AMSTEXT/48
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-5691-7
Product Code:  AMSTEXT/48.E
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Softcover ISBN:  978-1-4704-5275-9
eBook: ISBN:  978-1-4704-5691-7
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
Number Theory
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Number Theory
Róbert Freud University Eötvös Loránd, Budapest, Hungary
Softcover ISBN:  978-1-4704-5275-9
Product Code:  AMSTEXT/48
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-5691-7
Product Code:  AMSTEXT/48.E
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Softcover ISBN:  978-1-4704-5275-9
eBook ISBN:  978-1-4704-5691-7
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 Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 482020; 549 pp
    MSC: 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 well-known 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:

    Readership

    Undergraduate 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 482020; 549 pp
MSC: 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 well-known 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:

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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