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Number Theory Revealed: An Introduction
 
Andrew Granville University of Montreal, Quebec, Canada and University College London, London, England and (formerly) University of Georgia, Athens, GA
Front Cover for Number Theory Revealed: An Introduction
HardcoverISBN:  978-1-4704-4157-9
Product Code:  MBK/126
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBookISBN:  978-1-4704-5423-4
Product Code:  MBK/126.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
HardcoverISBN:  978-1-4704-4157-9
eBookISBN:  978-1-4704-5423-4
Product Code:  MBK/126.B
List Price: $138.00$103.50
MAA Member Price: $124.20$93.15
AMS Member Price: $110.40$82.80
Front Cover for Number Theory Revealed: An Introduction
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  • Front Cover for Number Theory Revealed: An Introduction
  • Back Cover for Number Theory Revealed: An Introduction
Number Theory Revealed: An Introduction
Andrew Granville University of Montreal, Quebec, Canada and University College London, London, England and (formerly) University of Georgia, Athens, GA
Hardcover ISBN:  978-1-4704-4157-9
Product Code:  MBK/126
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-5423-4
Product Code:  MBK/126.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
Hardcover ISBN:  978-1-4704-4157-9
eBookISBN:  978-1-4704-5423-4
Product Code:  MBK/126.B
List Price: $138.00$103.50
MAA Member Price: $124.20$93.15
AMS Member Price: $110.40$82.80
  • Book Details
     
     
    2019; 264 pp
    MSC: Primary 11;

    Number Theory Revealed: An Introduction presents a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations, as well as hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod p, Fermat's Last Theorem for polynomials, and modern twists on traditional questions. This book provides careful coverage of all core topics in a standard introductory number theory course with pointers to some exciting further directions.

    An expanded edition, Number Theory Revealed: A Masterclass, offers a more comprehensive approach, adding additional material in further chapters and appendices. It is ideal for instructors who wish to tailor a class to their own interests and gives well-prepared students further opportunities to challenge themselves and push beyond core number theory concepts, serving as a springboard to many current themes in mathematics.

    This book is part of Number Theory Revealed: The Series. Find full tables of contents, sample problems, hints, and appendices, as well as updates about forthcoming related volumes here.

    About the Author:

    Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. He has won several international writing prizes for exposition in mathematics, including the 2008 Chauvenet Prize and the 2019 Halmos-Ford Prize, and is the author of Prime Suspects (Princeton University Press, 2019), a beautifully illustrated graphic novel murder mystery that explores surprising connections between the anatomies of integers and of permutations.

    Readership

    Undergraduate and graduate students interested in introductory number theory.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Preface
    • Gauss’s Disquisitiones Arithmeticae
    • Notation
    • The language of mathematics
    • Prerequisites
    • Preliminary Chapter on Induction
    • 0.1. Fibonacci numbers and other recurrence sequences
    • 0.2. Formulas for sums of powers of integers
    • 0.3. The binomial theorem, Pascal’s triangle, and the binomial coefficients
    • Articles with further thoughts on factorials and binomial coefficients
    • Additional exercises
    • A paper that questions one’s assumptions is
    • Chapter 1. The Euclidean algorithm
    • 1.1. Finding the gcd
    • 1.2. Linear combinations
    • 1.3. The set of linear combinations of two integers
    • 1.4. The least common multiple
    • 1.5. Continued fractions
    • 1.6. Tiling a rectangle with squares
    • Additional exercises
    • Divisors in recurrence sequences
    • Appendix 1A. Reformulating the Euclidean algorithm
    • 1.8. Euclid matrices and Euclid’s algorithm
    • 1.9. Euclid matrices and ideal transformations
    • 1.10. The dynamics of the Euclidean algorithm
    • Chapter 2. Congruences
    • 2.1. Basic congruences
    • 2.2. The trouble with division
    • 2.3. Congruences for polynomials
    • 2.4. Tests for divisibility
    • Additional exercises
    • Binomial coefficients modulo 𝑝
    • The Fibonacci numbers modulo 𝑑
    • Appendix 2A. Congruences in the language of groups
    • 2.6. Further discussion of the basic notion of congruence
    • 2.7. Cosets of an additive group
    • 2.8. A new family of rings and fields
    • 2.9. The order of an element
    • Chapter 3. The basic algebra of number theory
    • 3.1. The Fundamental Theorem of Arithmetic
    • 3.2. Abstractions
    • 3.3. Divisors using factorizations
    • 3.4. Irrationality
    • 3.5. Dividing in congruences
    • 3.6. Linear equations in two unknowns
    • 3.7. Congruences to several moduli
    • 3.8. Square roots of 1 (mod 𝑛)
    • Additional exercises
    • Reference on the many proofs that √2 is irrational
    • Appendix 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝
    • 3.10. The prime powers dividing a given binomial coefficient
    • 3.11. Pascal’s triangle modulo 2
    • References for this chapter
    • Chapter 4. Multiplicative functions
    • 4.1. Euler’s 𝜙-function
    • 4.2. Perfect numbers. “The whole is equal to the sum of its parts.”
    • Additional exercises
    • Appendix 4A. More multiplicative functions
    • 4.4. Summing multiplicative functions
    • 4.5. Inclusion-exclusion and the Möbius function
    • 4.6. Convolutions and the Möbius inversion formula
    • 4.7. The Liouville function
    • Additional exercises
    • Chapter 5. The distribution of prime numbers
    • 5.1. Proofs that there are infinitely many primes
    • 5.2. Distinguishing primes
    • 5.3. Primes in certain arithmetic progressions
    • 5.4. How many primes are there up to 𝑥?
    • 5.5. Bounds on the number of primes
    • 5.6. Gaps between primes
    • Further reading on hot topics in this section
    • 5.7. Formulas for primes
    • Additional exercises
    • Appendix 5A. Bertrand’s postulate and beyond
    • 5.9. Bertrand’s postulate
    • 5.10. The theorem of Sylvester and Schur
    • Bonus read: A review of prime problems
    • 5.11. Prime problems
    • Prime values of polynomials in one variable
    • Prime values of polynomials in several variables
    • Goldbach’s conjecture and variants
    • Other questions
    • Guides to conjectures and the Green-Tao Theorem
    • Chapter 6. Diophantine problems
    • 6.1. The Pythagorean equation
    • 6.2. No solutions to a Diophantine equation through descent
    • No solutions through prime divisibility
    • No solutions through geometric descent
    • 6.3. Fermat’s “infinite descent”
    • 6.4. Fermat’s Last Theorem
    • A brief history of equation solving
    • References for this chapter
    • Additional exercises
    • Appendix 6A. Polynomial solutions of Diophantine equations
    • 6.6. Fermat’s Last Theorem in ℂ[𝕥]
    • 6.7. 𝑎+𝑏=𝑐 in ℂ[𝕥]
    • Chapter 7. Power residues
    • 7.1. Generating the multiplicative group of residues
    • 7.2. Fermat’s Little Theorem
    • 7.3. Special primes and orders
    • 7.4. Further observations
    • 7.5. The number of elements of a given order, and primitive roots
    • 7.6. Testing for composites, pseudoprimes, and Carmichael numbers
    • 7.7. Divisibility tests, again
    • 7.8. The decimal expansion of fractions
    • 7.9. Primes in arithmetic progressions, revisited
    • References for this chapter
    • Additional exercises
    • Appendix 7A. Card shuffling and Fermat’s Little Theorem
    • 7.11. Card shuffling and orders modulo 𝑛
    • 7.12. The “necklace proof” of Fermat’s Little Theorem
    • More combinatorics and number theory
    • 7.13. Taking powers efficiently
    • 7.14. Running time: The desirability of polynomial time algorithms
    • Chapter 8. Quadratic residues
    • 8.1. Squares modulo prime 𝑝
    • 8.2. The quadratic character of a residue
    • 8.3. The residue -1
    • 8.4. The residue 2
    • 8.5. The law of quadratic reciprocity
    • 8.6. Proof of the law of quadratic reciprocity
    • 8.7. The Jacobi symbol
    • 8.8. The squares modulo 𝑚
    • Additional exercises
    • Further reading on Euclidean proofs
    • Appendix 8A. Eisenstein’s proof of quadratic reciprocity
    • 8.10. Eisenstein’s elegant proof, 1844
    • Chapter 9. Quadratic equations
    • 9.1. Sums of two squares
    • 9.2. The values of 𝑥²+𝑑𝑦²
    • 9.3. Is there a solution to a given quadratic equation?
    • 9.4. Representation of integers by 𝑎𝑥²+𝑏𝑦² with 𝑥,𝑦 rational, and beyond
    • 9.5. The failure of the local-global principle for quadratic equations in integers
    • 9.6. Primes represented by 𝑥²+5𝑦²
    • Additional exercises
    • Appendix 9A. Proof of the local-global principle for quadratic equations
    • 9.8. Lattices and quotients
    • 9.9. A better proof of the local-global principle
    • Chapter 10. Square roots and factoring
    • 10.1. Square roots modulo 𝑛
    • 10.2. Cryptosystems
    • 10.3. RSA
    • 10.4. Certificates and the complexity classes P and NP
    • 10.5. Polynomial time primality testing
    • 10.6. Factoring methods
    • References: See [CP05] and [Knu98], as well as:
    • Additional exercises
    • Appendix 10A. Pseudoprime tests using square roots of 1
    • 10.8. The difficulty of finding all square roots of 1
    • Chapter 11. Rational approximations to real numbers
    • 11.1. The pigeonhole principle
    • 11.2. Pell’s equation
    • 11.3. Descent on solutions of 𝑥²-𝑑𝑦²=𝑛,𝑑>0
    • 11.4. Transcendental numbers
    • 11.5. The 𝑎𝑏𝑐-conjecture
    • Further reading for this chapter
    • Additional exercises
    • Appendix 11A. Uniform distribution
    • 11.7. 𝑛𝛼 mod 1
    • 11.8. Bouncing billiard balls
    • Chapter 12. Binary quadratic forms
    • 12.1. Representation of integers by binary quadratic forms
    • 12.2. Equivalence classes of binary quadratic forms
    • 12.3. Congruence restrictions on the values of a binary quadratic form
    • 12.4. Class numbers
    • 12.5. Class number one
    • References for this chapter
    • Additional exercises
    • Appendix 12A. Composition rules: Gauss, Dirichlet, and Bhargava
    • 12.7. Composition and Gauss
    • 12.8. Dirichlet composition
    • 12.9. Bhargava composition
    • Hints for exercises
    • Recommended further reading
    • Index
    • Back Cover
  • Reviews
     
     
    • In 'Number Theory Revealed: An Introduction,' Andrew Granville presents a fresh take on the classic structure of a number theory textbook. While it includes the standard topics that one would expect to find in a textbook on elementary number theory, it is also filled throughout with related problems, different approaches to proving key theorems, and interesting digressions that will be of interest even to more advanced readers. At the same time, it assumes relatively little background --- starting, for example, with an introductory chapter on induction --- making it accessible to a wide audience.

      Nathan McNew (Towson University), MAA Reviews
    • I strongly recommend the reading of 'Number Theory Revealed' (the 'Masterclass' in particular) not only to all mathematicians but also to anybody scientifically inclined and curious about what mathematics is and how it is done. Not only are the topics well chosen and well presented, but this book is a real page-turner. How often can you say that about a mathematical textbook? Chapeau!

      Marco Abate, The Mathematical Intelligencer
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    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
2019; 264 pp
MSC: Primary 11;

Number Theory Revealed: An Introduction presents a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations, as well as hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod p, Fermat's Last Theorem for polynomials, and modern twists on traditional questions. This book provides careful coverage of all core topics in a standard introductory number theory course with pointers to some exciting further directions.

An expanded edition, Number Theory Revealed: A Masterclass, offers a more comprehensive approach, adding additional material in further chapters and appendices. It is ideal for instructors who wish to tailor a class to their own interests and gives well-prepared students further opportunities to challenge themselves and push beyond core number theory concepts, serving as a springboard to many current themes in mathematics.

This book is part of Number Theory Revealed: The Series. Find full tables of contents, sample problems, hints, and appendices, as well as updates about forthcoming related volumes here.

About the Author:

Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. He has won several international writing prizes for exposition in mathematics, including the 2008 Chauvenet Prize and the 2019 Halmos-Ford Prize, and is the author of Prime Suspects (Princeton University Press, 2019), a beautifully illustrated graphic novel murder mystery that explores surprising connections between the anatomies of integers and of permutations.

Readership

Undergraduate and graduate students interested in introductory number theory.

  • Cover
  • Title page
  • Preface
  • Gauss’s Disquisitiones Arithmeticae
  • Notation
  • The language of mathematics
  • Prerequisites
  • Preliminary Chapter on Induction
  • 0.1. Fibonacci numbers and other recurrence sequences
  • 0.2. Formulas for sums of powers of integers
  • 0.3. The binomial theorem, Pascal’s triangle, and the binomial coefficients
  • Articles with further thoughts on factorials and binomial coefficients
  • Additional exercises
  • A paper that questions one’s assumptions is
  • Chapter 1. The Euclidean algorithm
  • 1.1. Finding the gcd
  • 1.2. Linear combinations
  • 1.3. The set of linear combinations of two integers
  • 1.4. The least common multiple
  • 1.5. Continued fractions
  • 1.6. Tiling a rectangle with squares
  • Additional exercises
  • Divisors in recurrence sequences
  • Appendix 1A. Reformulating the Euclidean algorithm
  • 1.8. Euclid matrices and Euclid’s algorithm
  • 1.9. Euclid matrices and ideal transformations
  • 1.10. The dynamics of the Euclidean algorithm
  • Chapter 2. Congruences
  • 2.1. Basic congruences
  • 2.2. The trouble with division
  • 2.3. Congruences for polynomials
  • 2.4. Tests for divisibility
  • Additional exercises
  • Binomial coefficients modulo 𝑝
  • The Fibonacci numbers modulo 𝑑
  • Appendix 2A. Congruences in the language of groups
  • 2.6. Further discussion of the basic notion of congruence
  • 2.7. Cosets of an additive group
  • 2.8. A new family of rings and fields
  • 2.9. The order of an element
  • Chapter 3. The basic algebra of number theory
  • 3.1. The Fundamental Theorem of Arithmetic
  • 3.2. Abstractions
  • 3.3. Divisors using factorizations
  • 3.4. Irrationality
  • 3.5. Dividing in congruences
  • 3.6. Linear equations in two unknowns
  • 3.7. Congruences to several moduli
  • 3.8. Square roots of 1 (mod 𝑛)
  • Additional exercises
  • Reference on the many proofs that √2 is irrational
  • Appendix 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝
  • 3.10. The prime powers dividing a given binomial coefficient
  • 3.11. Pascal’s triangle modulo 2
  • References for this chapter
  • Chapter 4. Multiplicative functions
  • 4.1. Euler’s 𝜙-function
  • 4.2. Perfect numbers. “The whole is equal to the sum of its parts.”
  • Additional exercises
  • Appendix 4A. More multiplicative functions
  • 4.4. Summing multiplicative functions
  • 4.5. Inclusion-exclusion and the Möbius function
  • 4.6. Convolutions and the Möbius inversion formula
  • 4.7. The Liouville function
  • Additional exercises
  • Chapter 5. The distribution of prime numbers
  • 5.1. Proofs that there are infinitely many primes
  • 5.2. Distinguishing primes
  • 5.3. Primes in certain arithmetic progressions
  • 5.4. How many primes are there up to 𝑥?
  • 5.5. Bounds on the number of primes
  • 5.6. Gaps between primes
  • Further reading on hot topics in this section
  • 5.7. Formulas for primes
  • Additional exercises
  • Appendix 5A. Bertrand’s postulate and beyond
  • 5.9. Bertrand’s postulate
  • 5.10. The theorem of Sylvester and Schur
  • Bonus read: A review of prime problems
  • 5.11. Prime problems
  • Prime values of polynomials in one variable
  • Prime values of polynomials in several variables
  • Goldbach’s conjecture and variants
  • Other questions
  • Guides to conjectures and the Green-Tao Theorem
  • Chapter 6. Diophantine problems
  • 6.1. The Pythagorean equation
  • 6.2. No solutions to a Diophantine equation through descent
  • No solutions through prime divisibility
  • No solutions through geometric descent
  • 6.3. Fermat’s “infinite descent”
  • 6.4. Fermat’s Last Theorem
  • A brief history of equation solving
  • References for this chapter
  • Additional exercises
  • Appendix 6A. Polynomial solutions of Diophantine equations
  • 6.6. Fermat’s Last Theorem in ℂ[𝕥]
  • 6.7. 𝑎+𝑏=𝑐 in ℂ[𝕥]
  • Chapter 7. Power residues
  • 7.1. Generating the multiplicative group of residues
  • 7.2. Fermat’s Little Theorem
  • 7.3. Special primes and orders
  • 7.4. Further observations
  • 7.5. The number of elements of a given order, and primitive roots
  • 7.6. Testing for composites, pseudoprimes, and Carmichael numbers
  • 7.7. Divisibility tests, again
  • 7.8. The decimal expansion of fractions
  • 7.9. Primes in arithmetic progressions, revisited
  • References for this chapter
  • Additional exercises
  • Appendix 7A. Card shuffling and Fermat’s Little Theorem
  • 7.11. Card shuffling and orders modulo 𝑛
  • 7.12. The “necklace proof” of Fermat’s Little Theorem
  • More combinatorics and number theory
  • 7.13. Taking powers efficiently
  • 7.14. Running time: The desirability of polynomial time algorithms
  • Chapter 8. Quadratic residues
  • 8.1. Squares modulo prime 𝑝
  • 8.2. The quadratic character of a residue
  • 8.3. The residue -1
  • 8.4. The residue 2
  • 8.5. The law of quadratic reciprocity
  • 8.6. Proof of the law of quadratic reciprocity
  • 8.7. The Jacobi symbol
  • 8.8. The squares modulo 𝑚
  • Additional exercises
  • Further reading on Euclidean proofs
  • Appendix 8A. Eisenstein’s proof of quadratic reciprocity
  • 8.10. Eisenstein’s elegant proof, 1844
  • Chapter 9. Quadratic equations
  • 9.1. Sums of two squares
  • 9.2. The values of 𝑥²+𝑑𝑦²
  • 9.3. Is there a solution to a given quadratic equation?
  • 9.4. Representation of integers by 𝑎𝑥²+𝑏𝑦² with 𝑥,𝑦 rational, and beyond
  • 9.5. The failure of the local-global principle for quadratic equations in integers
  • 9.6. Primes represented by 𝑥²+5𝑦²
  • Additional exercises
  • Appendix 9A. Proof of the local-global principle for quadratic equations
  • 9.8. Lattices and quotients
  • 9.9. A better proof of the local-global principle
  • Chapter 10. Square roots and factoring
  • 10.1. Square roots modulo 𝑛
  • 10.2. Cryptosystems
  • 10.3. RSA
  • 10.4. Certificates and the complexity classes P and NP
  • 10.5. Polynomial time primality testing
  • 10.6. Factoring methods
  • References: See [CP05] and [Knu98], as well as:
  • Additional exercises
  • Appendix 10A. Pseudoprime tests using square roots of 1
  • 10.8. The difficulty of finding all square roots of 1
  • Chapter 11. Rational approximations to real numbers
  • 11.1. The pigeonhole principle
  • 11.2. Pell’s equation
  • 11.3. Descent on solutions of 𝑥²-𝑑𝑦²=𝑛,𝑑>0
  • 11.4. Transcendental numbers
  • 11.5. The 𝑎𝑏𝑐-conjecture
  • Further reading for this chapter
  • Additional exercises
  • Appendix 11A. Uniform distribution
  • 11.7. 𝑛𝛼 mod 1
  • 11.8. Bouncing billiard balls
  • Chapter 12. Binary quadratic forms
  • 12.1. Representation of integers by binary quadratic forms
  • 12.2. Equivalence classes of binary quadratic forms
  • 12.3. Congruence restrictions on the values of a binary quadratic form
  • 12.4. Class numbers
  • 12.5. Class number one
  • References for this chapter
  • Additional exercises
  • Appendix 12A. Composition rules: Gauss, Dirichlet, and Bhargava
  • 12.7. Composition and Gauss
  • 12.8. Dirichlet composition
  • 12.9. Bhargava composition
  • Hints for exercises
  • Recommended further reading
  • Index
  • Back Cover
  • In 'Number Theory Revealed: An Introduction,' Andrew Granville presents a fresh take on the classic structure of a number theory textbook. While it includes the standard topics that one would expect to find in a textbook on elementary number theory, it is also filled throughout with related problems, different approaches to proving key theorems, and interesting digressions that will be of interest even to more advanced readers. At the same time, it assumes relatively little background --- starting, for example, with an introductory chapter on induction --- making it accessible to a wide audience.

    Nathan McNew (Towson University), MAA Reviews
  • I strongly recommend the reading of 'Number Theory Revealed' (the 'Masterclass' in particular) not only to all mathematicians but also to anybody scientifically inclined and curious about what mathematics is and how it is done. Not only are the topics well chosen and well presented, but this book is a real page-turner. How often can you say that about a mathematical textbook? Chapeau!

    Marco Abate, The Mathematical Intelligencer
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
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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