2019;
587 pp;
Hardcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-4158-6
Product Code: MBK/127**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-5424-1
DOI: 10.1090/MBK/127
Product Code: MBK/127.E**

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AMS Member Price: $79.20

MAA Member Price: $89.10

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#### Supplemental Materials

# Number Theory Revealed: A Masterclass

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*Andrew Granville*

Number Theory Revealed: A Masterclass presents a
fresh take on congruences, power residues, quadratic residues, primes,
and Diophantine equations and presents 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 Masterclass edition contains many additional chapters
and appendices not found in Number Theory Revealed: An
Introduction. 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. Additional topics in A Masterclass include the curvature
of circles in a tiling of a circle by circles, the latest discoveries
on gaps between primes, magic squares of primes, a new proof of
Mordell's Theorem for congruent elliptic curves, as well as links with
algebra, analysis, cryptography, and dynamics.

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.

#### Reviews & Endorsements

Andrew Granville has written many wonderful expository articles about number theory (especially patterns in primes), as well as dozens of research articles on many aspects of this field. It is, thus, highly appropriate that he turn all this knowledge into a series of textbooks for number theory. So if you are looking for enrichment in how the world of number theory connects together but formatted as a textbook with familiar topical arrangement, this is a great resource; I can't wait for the next two volumes to appear.

-- Karl-Dieter Crisman, MAA Reviews

#### Table of Contents

# Table of Contents

## Number Theory Revealed: A Masterclass

Table of Contents pages: 1 2 3

- Title page 44
- Preface 1818
- Gauss’s Disquisitiones Arithmeticae 2424
- Notation 2626
- Prerequisites 2828
- Preliminary Chapter on Induction 3030
- 0.1. Fibonacci numbers and other recurrence sequences 3030
- 0.2. Formulas for sums of powers of integers 3232
- 0.3. The binomial theorem, Pascal’s triangle, and the binomial coefficients 3333
- Articles with further thoughts on factorials and binomial coefficients 3535
- Additional exercises 3535
- A paper that questions one’s assumptions is 3737
- 0A. A closed formula for sums of powers 3838
- 0.5. Formulas for sums of powers of integers, II 3838
- 0B. Generating functions 4040
- 0.6. Formulas for sums of powers of integers, III 4040
- 0.7. The power series view on the Fibonacci numbers 4242
- 0C. Finding roots of polynomials 4444
- 0.8. Solving the general cubic 4444
- 0.9. Solving the general quartic 4545
- 0.10. Surds 4646
- References discussing solvability of polynomials 4747
- 0D. What is a group? 4848
- 0.11. Examples and definitions 4848
- 0.12. Matrices usually don’t commute 4949
- 0E. Rings and fields 5151
- 0.13. Mixing addition and multiplication together: Rings and fields 5151
- 0.14. Algebraic numbers, integers, and units, I 5252
- 0F. Symmetric polynomials 5454
- 0.15. The theory of symmetric polynomials 5454
- 0.16. Some special symmetric polynomials 5656
- 0.17. Algebraic numbers, integers, and units, II 5858
- 0G. Constructibility 5959
- 0.18. Constructible using only compass and ruler 5959

- Chapter 1. The Euclidean algorithm 6262
- 1.1. Finding the gcd 6262
- 1.2. Linear combinations 6464
- 1.3. The set of linear combinations of two integers 6666
- 1.4. The least common multiple 6868
- 1.5. Continued fractions 6868
- 1.6. Tiling a rectangle with squares 7070
- Additional exercises 7171
- Divisors in recurrence sequences 7373
- 1A. Reformulating the Euclidean algorithm 7474
- 1.8. Euclid matrices and Euclid’s algorithm 7474
- 1.9. Euclid matrices and ideal transformations 7676
- 1.10. The dynamics of the Euclidean algorithm 7777
- 1B. Computational aspects of the Euclidean algorithm 8080
- 1.11. Speeding up the Euclidean algorithm 8080
- 1.12. Euclid’s algorithm works in “polynomial time” 8181
- 1C. Magic squares 8383
- 1.13. Turtle power 8383
- 1.14. Latin squares 8484
- 1.15. Factoring magic squares 8585
- Reference for this appendix 8585
- 1D. The Frobenius postage stamp problem 8686
- 1.16. The Frobenius postage stamp problem, I 8686
- 1E. Egyptian fractions 8888
- 1.17. Simple fractions 8888

- Chapter 2. Congruences 9090
- 2.1. Basic congruences 9090
- 2.2. The trouble with division 9393
- 2.3. Congruences for polynomials 9595
- 2.4. Tests for divisibility 9595
- Additional exercises 9696
- Binomial coefficients modulo 𝑝 9797
- The Fibonacci numbers modulo 𝑑 9898
- 2A. Congruences in the language of groups 100100
- 2.6. Further discussion of the basic notion of congruence 100100
- 2.7. Cosets of an additive group 101101
- 2.8. A new family of rings and fields 102102
- 2.9. The order of an element 102102
- 2B. The Euclidean algorithm for polynomials 104104
- 2.10. The Euclidean algorithm in ℂ[𝕩] 104104
- 2.11. Common factors over rings: Resultants and discriminants 106106
- 2.12. Euclidean domains 107107

- Chapter 3. The basic algebra of number theory 110110
- 3.1. The Fundamental Theorem of Arithmetic 110110
- 3.2. Abstractions 112112
- 3.3. Divisors using factorizations 114114
- 3.4. Irrationality 116116
- 3.5. Dividing in congruences 117117
- 3.6. Linear equations in two unknowns 119119
- 3.7. Congruences to several moduli 121121
- 3.8. Square roots of 1 (mod 𝑛) 123123
- Additional exercises 125125
- Reference on the many proofs that √2 is irrational 126126
- 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝 128128
- 3.10. The prime powers dividing a given binomial coefficient 128128
- 3.11. Pascal’s triangle modulo 2 130130
- References for this chapter 132132
- 3B. Solving linear congruences 133133
- 3.12. Composite moduli 133133
- 3.13. Solving linear congruences with several unknowns 134134
- 3.14. The Chinese Remainder Theorem in general 135135
- When the moduli are not coprime 135135
- 3C. Groups and rings 138138
- 3.15. A direct sum 138138
- 3.16. The structure of finite abelian groups 139139
- 3D. Unique factorization revisited 141141
- 3.17. The Fundamental Theorem of Arithmetic, clarified 141141
- 3.18. When unique factorization fails 142142
- 3.19. Defining ideals and factoring 142142
- 3.20. Bases for ideals in quadratic fields 144144
- 3E. Gauss’s approach 145145
- 3.21. Gauss’s approach to Euclid’s Lemma 145145
- 3F. Fundamental theorems and factoring polynomials 146146
- 3.22. The number of distinct roots of polynomials 146146
- The Euclidean algorithm for polynomials 147147
- Unique factorization of polynomials modulo 𝑝 149149
- 3.23. Interpreting resultants and discriminants 149149
- 3.24. Other approaches to resultants and gcds 150150
- Additional exercises 151151
- 3G. Open problems 152152
- 3.25. The Frobenius postage stamp problem, II 152152
- 3.26. Egyptian fractions for 3/𝑏 153153
- 3.27. The 3𝑥+1 conjecture 153153
- Further reading on these open problems 155155

- Chapter 4. Multiplicative functions 156156
- 4.1. Euler’s 𝜙-function 157157
- 4.2. Perfect numbers. “The whole is equal to the sum of its parts.” 158158
- Additional exercises 160160
- 4A. More multiplicative functions 163163
- 4.4. Summing multiplicative functions 163163
- 4.5. Inclusion-exclusion and the Möbius function 164164
- 4.6. Convolutions and the Möbius inversion formula 165165
- 4.7. The Liouville function 167167
- Additional exercises 168168
- 4B. Dirichlet series and multiplicative functions 169169
- 4.9. Dirichlet series 169169
- 4.10. Multiplication of Dirichlet series 170170
- 4.11. Other Dirichlet series of interest 171171
- 4C. Irreducible polynomials modulo 𝑝 173173
- 4.12. Irreducible polynomials modulo 𝑝 173173
- 4D. The harmonic sum and the divisor function 176176
- 4.13. The average number of divisors 176176
- 4.14. The harmonic sum 177177
- 4.15. Dirichlet’s hyperbola trick 179179
- 4E. Cyclotomic polynomials 182182
- 4.16. Cyclotomic polynomials 182182

- Chapter 5. The distribution of prime numbers 184184
- 5.1. Proofs that there are infinitely many primes 184184
- 5.2. Distinguishing primes 186186
- 5.3. Primes in certain arithmetic progressions 188188
- 5.4. How many primes are there up to 𝑥? 189189
- 5.5. Bounds on the number of primes 192192
- 5.6. Gaps between primes 194194
- Further reading on hot topics in this section 196196
- 5.7. Formulas for primes 196196
- Additional exercises 198198
- 5A. Bertrand’s postulate and beyond 200200
- 5.9. Bertrand’s postulate 200200
- 5.10. The theorem of Sylvester and Schur 201201
- Bonus read: A review of prime problems 204204
- 5.11. Prime problems 204204
- Prime values of polynomials in one variable 204204
- Prime values of polynomials in several variables 206206
- Goldbach’s conjecture and variants 208208
- Other questions 209209
- Guides to conjectures and the Green-Tao Theorem 209209
- 5B. An important proof of infinitely many primes 211211
- 5.12. Euler’s proof of the infinitude of primes 211211
- Reference on Euler’s many contributions 214214
- 5.13. The sieve of Eratosthenes and estimates for the primes up to 𝑥 214214
- 5.14. Riemann’s plan for Gauss’s prediction, I 215215
- 5C. What should be true about primes? 216216
- 5.15. The Gauss-Cramér model for the primes 216216
- Short intervals 218218
- Twin primes 219219
- 5D. Working with Riemann’s zeta-function 221221
- 5.16. Riemann’s plan for Gauss’s prediction 221221
- 5.17. Understanding the zeros 223223
- An elementary proof 225225
- 5.18. Reformulations of the Riemann Hypothesis 226226
- Primes and complex analysis 226226
- 5E. Prime patterns: Consequences of the Green-Tao Theorem 227227
- 5.19. Generalized arithmetic progressions of primes 228228
- Consecutive prime values of a polynomial 229229
- Magic squares of primes 230230
- Primes as averages 230230
- 5F. A panoply of prime proofs 231231
- Furstenberg’s (point-set) topological proof 231231
- An analytic proof 232232
- A proof by irrationality 232232
- 5G. Searching for primes and prime formulas 233233
- 5.21. Searching for prime formulas 233233
- 5.22. Conway’s prime producing machine 234234
- 5.23. Ulam’s spiral 235235
- 5.24. Mills’s formula 236236
- Further reading on primes in surprising places 236236
- 5H. Dynamical systems and infinitely many primes 237237
- 5.25. A simpler formulation 237237
- 5.26. Different starting points 238238
- 5.27. Dynamical systems and the infinitude of primes 239239
- 5.28. Polynomial maps for which 0 is strictly preperiodic 240240
- References for this chapter 242242

- Chapter 6. Diophantine problems 244244