2019;
587 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-6370-0
Product Code: MBK/127.S**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

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

List Price: $99.00

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

Not just a list of results and definitions that were hopefully explained in class, this a book for sitting down and engaging with, pen in hand, ready for the exercises interlaced with the exposition. Students learn not just by seeing examples and special cases worked out in advance of a general statement, but by working things out for themselves...One striking feature of this book (and perhaps one argument for a stream of new books on old subjects) is its inclusion of current research in its additional topics...This book will surely address the precocious student.s desire to know what number theory is about now.

-- Patrick Ingram, York University, CMS Notes

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

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

- Cover Cover11
- Title page 34
- Preface 1718
- Gauss’s Disquisitiones Arithmeticae 2324
- Notation 2526
- Prerequisites 2728
- Preliminary Chapter on Induction 2930
- 0.1. Fibonacci numbers and other recurrence sequences 2930
- 0.2. Formulas for sums of powers of integers 3132
- 0.3. The binomial theorem, Pascal’s triangle, and the binomial coefficients 3233
- Articles with further thoughts on factorials and binomial coefficients 3435
- Additional exercises 3435
- A paper that questions one’s assumptions is 3637
- 0A. A closed formula for sums of powers 3738
- 0.5. Formulas for sums of powers of integers, II 3738
- 0B. Generating functions 3940
- 0.6. Formulas for sums of powers of integers, III 3940
- 0.7. The power series view on the Fibonacci numbers 4142
- 0C. Finding roots of polynomials 4344
- 0.8. Solving the general cubic 4344
- 0.9. Solving the general quartic 4445
- 0.10. Surds 4546
- References discussing solvability of polynomials 4647
- 0D. What is a group? 4748
- 0.11. Examples and definitions 4748
- 0.12. Matrices usually don’t commute 4849
- 0E. Rings and fields 5051
- 0.13. Mixing addition and multiplication together: Rings and fields 5051
- 0.14. Algebraic numbers, integers, and units, I 5152
- 0F. Symmetric polynomials 5354
- 0.15. The theory of symmetric polynomials 5354
- 0.16. Some special symmetric polynomials 5556
- 0.17. Algebraic numbers, integers, and units, II 5758
- 0G. Constructibility 5859
- 0.18. Constructible using only compass and ruler 5859

- Chapter 1. The Euclidean algorithm 6162
- 1.1. Finding the gcd 6162
- 1.2. Linear combinations 6364
- 1.3. The set of linear combinations of two integers 6566
- 1.4. The least common multiple 6768
- 1.5. Continued fractions 6768
- 1.6. Tiling a rectangle with squares 6970
- Additional exercises 7071
- Divisors in recurrence sequences 7273
- 1A. Reformulating the Euclidean algorithm 7374
- 1.8. Euclid matrices and Euclid’s algorithm 7374
- 1.9. Euclid matrices and ideal transformations 7576
- 1.10. The dynamics of the Euclidean algorithm 7677
- 1B. Computational aspects of the Euclidean algorithm 7980
- 1.11. Speeding up the Euclidean algorithm 7980
- 1.12. Euclid’s algorithm works in “polynomial time” 8081
- 1C. Magic squares 8283
- 1.13. Turtle power 8283
- 1.14. Latin squares 8384
- 1.15. Factoring magic squares 8485
- Reference for this appendix 8485
- 1D. The Frobenius postage stamp problem 8586
- 1.16. The Frobenius postage stamp problem, I 8586
- 1E. Egyptian fractions 8788
- 1.17. Simple fractions 8788

- Chapter 2. Congruences 8990
- 2.1. Basic congruences 8990
- 2.2. The trouble with division 9293
- 2.3. Congruences for polynomials 9495
- 2.4. Tests for divisibility 9495
- Additional exercises 9596
- Binomial coefficients modulo 𝑝 9697
- The Fibonacci numbers modulo 𝑑 9798
- 2A. Congruences in the language of groups 99100
- 2.6. Further discussion of the basic notion of congruence 99100
- 2.7. Cosets of an additive group 100101
- 2.8. A new family of rings and fields 101102
- 2.9. The order of an element 101102
- 2B. The Euclidean algorithm for polynomials 103104
- 2.10. The Euclidean algorithm in ℂ[𝕩] 103104
- 2.11. Common factors over rings: Resultants and discriminants 105106
- 2.12. Euclidean domains 106107

- Chapter 3. The basic algebra of number theory 109110
- 3.1. The Fundamental Theorem of Arithmetic 109110
- 3.2. Abstractions 111112
- 3.3. Divisors using factorizations 113114
- 3.4. Irrationality 115116
- 3.5. Dividing in congruences 116117
- 3.6. Linear equations in two unknowns 118119
- 3.7. Congruences to several moduli 120121
- 3.8. Square roots of 1 (mod 𝑛) 122123
- Additional exercises 124125
- Reference on the many proofs that √2 is irrational 125126
- 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝 127128
- 3.10. The prime powers dividing a given binomial coefficient 127128
- 3.11. Pascal’s triangle modulo 2 129130
- References for this chapter 131132
- 3B. Solving linear congruences 132133
- 3.12. Composite moduli 132133
- 3.13. Solving linear congruences with several unknowns 133134
- 3.14. The Chinese Remainder Theorem in general 134135
- When the moduli are not coprime 134135
- 3C. Groups and rings 137138
- 3.15. A direct sum 137138
- 3.16. The structure of finite abelian groups 138139
- 3D. Unique factorization revisited 140141
- 3.17. The Fundamental Theorem of Arithmetic, clarified 140141
- 3.18. When unique factorization fails 141142
- 3.19. Defining ideals and factoring 141142
- 3.20. Bases for ideals in quadratic fields 143144
- 3E. Gauss’s approach 144145
- 3.21. Gauss’s approach to Euclid’s Lemma 144145
- 3F. Fundamental theorems and factoring polynomials 145146
- 3.22. The number of distinct roots of polynomials 145146
- The Euclidean algorithm for polynomials 146147
- Unique factorization of polynomials modulo 𝑝 148149
- 3.23. Interpreting resultants and discriminants 148149
- 3.24. Other approaches to resultants and gcds 149150
- Additional exercises 150151
- 3G. Open problems 151152
- 3.25. The Frobenius postage stamp problem, II 151152
- 3.26. Egyptian fractions for 3/𝑏 152153
- 3.27. The 3𝑥+1 conjecture 152153
- Further reading on these open problems 154155

- Chapter 4. Multiplicative functions 155156
- 4.1. Euler’s 𝜙-function 156157
- 4.2. Perfect numbers. “The whole is equal to the sum of its parts.” 157158
- Additional exercises 159160
- 4A. More multiplicative functions 162163
- 4.4. Summing multiplicative functions 162163
- 4.5. Inclusion-exclusion and the Möbius function 163164
- 4.6. Convolutions and the Möbius inversion formula 164165
- 4.7. The Liouville function 166167
- Additional exercises 167168
- 4B. Dirichlet series and multiplicative functions 168169
- 4.9. Dirichlet series 168169
- 4.10. Multiplication of Dirichlet series 169170
- 4.11. Other Dirichlet series of interest 170171
- 4C. Irreducible polynomials modulo 𝑝 172173
- 4.12. Irreducible polynomials modulo 𝑝 172173
- 4D. The harmonic sum and the divisor function 175176
- 4.13. The average number of divisors 175176
- 4.14. The harmonic sum 176177
- 4.15. Dirichlet’s hyperbola trick 178179
- 4E. Cyclotomic polynomials 181182
- 4.16. Cyclotomic polynomials 181182

- Chapter 5. The distribution of prime numbers 183184
- 5.1. Proofs that there are infinitely many primes 183184
- 5.2. Distinguishing primes 185186
- 5.3. Primes in certain arithmetic progressions 187188
- 5.4. How many primes are there up to 𝑥? 188189
- 5.5. Bounds on the number of primes 191192
- 5.6. Gaps between primes 193194
- Further reading on hot topics in this section 195196
- 5.7. Formulas for primes 195196
- Additional exercises 197198
- 5A. Bertrand’s postulate and beyond 199200
- 5.9. Bertrand’s postulate 199200
- 5.10. The theorem of Sylvester and Schur 200201
- Bonus read: A review of prime problems 203204
- 5.11. Prime problems 203204
- Prime values of polynomials in one variable 203204
- Prime values of polynomials in several variables 205206
- Goldbach’s conjecture and variants 207208
- Other questions 208209
- Guides to conjectures and the Green-Tao Theorem 208209
- 5B. An important proof of infinitely many primes 210211
- 5.12. Euler’s proof of the infinitude of primes 210211
- Reference on Euler’s many contributions 213214
- 5.13. The sieve of Eratosthenes and estimates for the primes up to 𝑥 213214
- 5.14. Riemann’s plan for Gauss’s prediction, I 214215
- 5C. What should be true about primes? 215216
- 5.15. The Gauss-Cramér model for the primes 215216
- Short intervals 217218
- Twin primes 218219
- 5D. Working with Riemann’s zeta-function 220221
- 5.16. Riemann’s plan for Gauss’s prediction 220221
- 5.17. Understanding the zeros 222223
- An elementary proof 224225
- 5.18. Reformulations of the Riemann Hypothesis 225226
- Primes and complex analysis 225226
- 5E. Prime patterns: Consequences of the Green-Tao Theorem 226227
- 5.19. Generalized arithmetic progressions of primes 227228
- Consecutive prime values of a polynomial 228229
- Magic squares of primes 229230
- Primes as averages 229230
- 5F. A panoply of prime proofs 230231
- Furstenberg’s (point-set) topological proof 230231
- An analytic proof 231232
- A proof by irrationality 231232
- 5G. Searching for primes and prime formulas 232233
- 5.21. Searching for prime formulas 232233
- 5.22. Conway’s prime producing machine 233234
- 5.23. Ulam’s spiral 234235
- 5.24. Mills’s formula 235236
- Further reading on primes in surprising places 235236
- 5H. Dynamical systems and infinitely many primes 236237
- 5.25. A simpler formulation 236237
- 5.26. Different starting points 237238
- 5.27. Dynamical systems and the infinitude of primes 238239
- 5.28. Polynomial maps for which 0 is strictly preperiodic 239240
- References for this chapter 241242

- Chapter 6. Diophantine problems 243244