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Hardcover ISBN:  9781470441579 
Product Code:  MBK/126 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $63.20 
Sale Price:  $51.35 
eBook ISBN:  9781470454234 
Product Code:  MBK/126.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $60.00 
Sale Price:  $48.75 
Hardcover ISBN:  9781470441579 
eBook ISBN:  9781470454234 
Product Code:  MBK/126.B 
List Price:  $154.00 $116.50 
MAA Member Price:  $138.60 $104.85 
AMS Member Price:  $123.20 $93.20 
Sale Price:  $100.10 $75.73 

Book Details2019; 264 ppMSC: 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 wellprepared 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 HalmosFord 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.
ReadershipUndergraduate 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 𝜙function

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. Inclusionexclusion 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 GreenTao 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 localglobal principle for quadratic equations in integers

9.6. Primes represented by 𝑥²+5𝑦²

Additional exercises

Appendix 9A. Proof of the localglobal principle for quadratic equations

9.8. Lattices and quotients

9.9. A better proof of the localglobal 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 𝑥²𝑑𝑦²=𝑛,𝑑>0

11.4. Transcendental numbers

11.5. The 𝑎𝑏𝑐conjecture

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


Additional Material

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 pageturner. How often can you say that about a mathematical textbook? Chapeau!
Marco Abate, The Mathematical Intelligencer


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 Book Details
 Table of Contents
 Additional Material
 Reviews
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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 wellprepared 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 HalmosFord 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.
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 𝜙function

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. Inclusionexclusion 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 GreenTao 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 localglobal principle for quadratic equations in integers

9.6. Primes represented by 𝑥²+5𝑦²

Additional exercises

Appendix 9A. Proof of the localglobal principle for quadratic equations

9.8. Lattices and quotients

9.9. A better proof of the localglobal 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 𝑥²𝑑𝑦²=𝑛,𝑑>0

11.4. Transcendental numbers

11.5. The 𝑎𝑏𝑐conjecture

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 pageturner. How often can you say that about a mathematical textbook? Chapeau!
Marco Abate, The Mathematical Intelligencer