Number Theory Revealed: A Masterclass

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Number Theory Revealed: A Masterclass

Andrew Granville : University of Montreal, Quebec, Canada and University College London, London, England and (formerly) University of Georgia, Athens, GA

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

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

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Softcover ISBN:	978-1-4704-6370-0
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Number Theory Revealed: A Masterclass

Andrew Granville : University of Montreal, Quebec, Canada and University College London, London, England and (formerly) University of Georgia, Athens, GA

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

List Price:	$99.00
MAA Member Price:	$89.10
AMS Member Price:	$79.20
Sale Price:	$64.35

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

List Price:	$89.00
MAA Member Price:	$80.10
AMS Member Price:	$71.20
Sale Price:	$57.85

Softcover ISBN:	978-1-4704-6370-0
eBook ISBN:	978-1-4704-5424-1
Product Code:	MBK/127.S.B

List Price:	$188.00 $143.50
MAA Member Price:	$169.20 $129.15
AMS Member Price:	$150.40 $114.80
Sale Price:	$122.20 $93.28

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2019; 587 pp

MSC: Primary 11

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.

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
0A. A closed formula for sums of powers
0.5. Formulas for sums of powers of integers, II
0B. Generating functions
0.6. Formulas for sums of powers of integers, III
0.7. The power series view on the Fibonacci numbers
0C. Finding roots of polynomials
0.8. Solving the general cubic
0.9. Solving the general quartic
0.10. Surds
References discussing solvability of polynomials
0D. What is a group?
0.11. Examples and definitions
0.12. Matrices usually don’t commute
0E. Rings and fields
0.13. Mixing addition and multiplication together: Rings and fields
0.14. Algebraic numbers, integers, and units, I
0F. Symmetric polynomials
0.15. The theory of symmetric polynomials
0.16. Some special symmetric polynomials
0.17. Algebraic numbers, integers, and units, II
0G. Constructibility
0.18. Constructible using only compass and ruler
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
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
1B. Computational aspects of the Euclidean algorithm
1.11. Speeding up the Euclidean algorithm
1.12. Euclid’s algorithm works in “polynomial time”
1C. Magic squares
1.13. Turtle power
1.14. Latin squares
1.15. Factoring magic squares
Reference for this appendix
1D. The Frobenius postage stamp problem
1.16. The Frobenius postage stamp problem, I
1E. Egyptian fractions
1.17. Simple fractions
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
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
2B. The Euclidean algorithm for polynomials
2.10. The Euclidean algorithm in ℂ[𝕩]
2.11. Common factors over rings: Resultants and discriminants
2.12. Euclidean domains
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
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
3B. Solving linear congruences
3.12. Composite moduli
3.13. Solving linear congruences with several unknowns
3.14. The Chinese Remainder Theorem in general
When the moduli are not coprime
3C. Groups and rings
3.15. A direct sum
3.16. The structure of finite abelian groups
3D. Unique factorization revisited
3.17. The Fundamental Theorem of Arithmetic, clarified
3.18. When unique factorization fails
3.19. Defining ideals and factoring
3.20. Bases for ideals in quadratic fields
3E. Gauss’s approach
3.21. Gauss’s approach to Euclid’s Lemma
3F. Fundamental theorems and factoring polynomials
3.22. The number of distinct roots of polynomials
The Euclidean algorithm for polynomials
Unique factorization of polynomials modulo 𝑝
3.23. Interpreting resultants and discriminants
3.24. Other approaches to resultants and gcds
Additional exercises
3G. Open problems
3.25. The Frobenius postage stamp problem, II
3.26. Egyptian fractions for 3/𝑏
3.27. The 3𝑥+1 conjecture
Further reading on these open problems
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
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
4B. Dirichlet series and multiplicative functions
4.9. Dirichlet series
4.10. Multiplication of Dirichlet series
4.11. Other Dirichlet series of interest
4C. Irreducible polynomials modulo 𝑝
4.12. Irreducible polynomials modulo 𝑝
4D. The harmonic sum and the divisor function
4.13. The average number of divisors
4.14. The harmonic sum
4.15. Dirichlet’s hyperbola trick
4E. Cyclotomic polynomials
4.16. Cyclotomic polynomials
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
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
5B. An important proof of infinitely many primes
5.12. Euler’s proof of the infinitude of primes
Reference on Euler’s many contributions
5.13. The sieve of Eratosthenes and estimates for the primes up to 𝑥
5.14. Riemann’s plan for Gauss’s prediction, I
5C. What should be true about primes?
5.15. The Gauss-Cramér model for the primes
Short intervals
Twin primes
5D. Working with Riemann’s zeta-function
5.16. Riemann’s plan for Gauss’s prediction
5.17. Understanding the zeros
An elementary proof
5.18. Reformulations of the Riemann Hypothesis
Primes and complex analysis
5E. Prime patterns: Consequences of the Green-Tao Theorem
5.19. Generalized arithmetic progressions of primes
Consecutive prime values of a polynomial
Magic squares of primes
Primes as averages
5F. A panoply of prime proofs
Furstenberg’s (point-set) topological proof
An analytic proof
A proof by irrationality
5G. Searching for primes and prime formulas
5.21. Searching for prime formulas
5.22. Conway’s prime producing machine
5.23. Ulam’s spiral
5.24. Mills’s formula
Further reading on primes in surprising places
5H. Dynamical systems and infinitely many primes
5.25. A simpler formulation
5.26. Different starting points
5.27. Dynamical systems and the infinitude of primes
5.28. Polynomial maps for which 0 is strictly preperiodic
References for this chapter
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
6A. Polynomial solutions of Diophantine equations
6.6. Fermat’s Last Theorem in ℂ[𝕥]
6.7. 𝑎+𝑏=𝑐 in ℂ[𝕥]
6B. No Pythagorean triangle of square area via Euclidean geometry
An algebraic proof, by descent
6.8. A geometric viewpoint
6C. Can a binomial coefficient be a square?
6.9. Small 𝑘
6.10. Larger 𝑘
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
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
7B. Orders and primitive roots
7.15. Constructing primitive roots modulo 𝑝
7.16. Indices / Discrete Logarithms
7.17. Primitive roots modulo prime powers
7.18. Orders modulo composites
7C. Finding 𝑛th roots modulo prime powers
7.19. 𝑛th roots modulo 𝑝
7.20. Lifting solutions
7.21. Finding 𝑛th roots quickly
7D. Orders for finite groups
7.22. Cosets of general groups
7.23. Lagrange and Wilson
7.24. Normal subgroups
7E. Constructing finite fields
7.25. Classification of finite fields
Further reading on arithmetic associated with finite fields
7.26. The product of linear forms in 𝔽_{𝕢}
7F. Sophie Germain and Fermat’s Last Theorem
7.27. Fermat’s Last Theorem and Sophie Germain
7G. Primes of the form 2ⁿ+𝑘
7.28. Covering sets of congruences
7.29. Covering systems for the Fibonacci numbers
A Fibonacci covering system
7.30. The theory of covering systems
Interesting articles on covering systems
7H. Further congruences
7.31. Fermat quotients
Binomial coefficients
Bernoulli numbers modulo 𝑝
Sums of powers of integers modulo 𝑝²
The Wilson quotient
Beyond Fermat’s Little Theorem
Reference for this section
7.32. Frequency of 𝑝-divisibility
Fermat quotients
Bernoulli numbers
7I. Primitive prime factors of recurrence sequences
7.33. Primitive prime factors
Prime power divisibility of second-order linear recurrence sequences
7.34. Closed form identities and sums of powers
7.35. Primitive prime factors and second-order linear recurrence sequences
References for this chapter
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
8A. Eisenstein’s proof of quadratic reciprocity
8.10. Eisenstein’s elegant proof, 1844
8B. Small quadratic non-residues
8.11. The least quadratic non-residue modulo 𝑝
8.12. The smallest prime 𝑞 for which 𝑝 is a quadratic non-residue modulo 𝑞
8.13. Character sums and the least quadratic non-residue
8C. The first proof of quadratic reciprocity
8.14. Gauss’s original proof of the law of quadratic reciprocity
8D. Dirichlet characters and primes in arithmetic progressions
8.15. The Legendre symbol and a certain quotient group
8.16. Dirichlet characters
8.17. Dirichlet series and primes in arithmetic progressions
Uniformity questions
8E. Quadratic reciprocity and recurrence sequences
8.18. The Fibonacci numbers modulo 𝑝
8.19. General second-order linear recurrence sequences modulo 𝑝
8.20. Prime values in recurrence sequences
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
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
9B. Reformulation of the local-global principle
9.10. The Hilbert symbol
9.11. The Hasse-Minkowski principle
This is all discussed in detail in Part I of the wonderful book
9C. The number of representations
9.12. Distinct representations as sums of two squares
9D. Descent and the quadratics
9.13. Further solutions through linear algebra
9.14. The Markov equation
9.15. Apollonian circle packing
Further reading on Apollonian packings
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
10A. Pseudoprime tests using square roots of 1
10.8. The difficulty of finding all square roots of 1
10B. Factoring with squares
Random squares
Euler’s sum of squares method
The Continued fractions method
10.9. Factoring with polynomial values
The large prime variation
10C. Identifying primes of a given size
10.10. The Proth-Pocklington-Lehmer primality test
Second-order linear recurrences
References for this chapter
10D. Carmichael numbers
10.11. Constructing Carmichael numbers
10.12. Erdős’s construction
The computational evidence
References for this chapter
10E. Cryptosystems based on discrete logarithms
10.13. The Diffie-Hellman key exchange
10.14. The El Gamal cryptosystem
10F. Running times of algorithms
10.15. P and NP
10.16. Difficult problems
10G. The AKS test
10.17. A computationally quicker characterization of the primes
10.18. A set of extraordinary congruences
Upper bounds on |𝐺|
Lower bounds on |𝐺|
References for this chapter
10H. Factoring algorithms for polynomials
10.19. Testing polynomials for irreducibility
10.20. Testing whether a polynomial is squarefree
10.21. Factoring a squarefree polynomial modulo 𝑝
References for this chapter
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
11A. Uniform distribution
11.7. 𝑛𝛼 mod 1
11.8. Bouncing billiard balls
11B. Continued fractions
11.9. Continued fractions for real numbers
11.10. How good are these approximations?
11.11. Periodic continued fractions and Pell’s equation
11.12. Quadratic irrationals and periodic continued fractions
11.13. Solutions to Pell’s equation from a well-selected continued fraction
11.14. Sums of two squares from continued fractions
Modern uses of continued fractions
11C. Two-variable quadratic equations
11.15. Integer solutions to 2-variable quadratics
11D. Transcendental numbers
11.16. Diagonalization
11.17. The hunt for transcendental numbers
11.18. Normal numbers
Normal digits of numbers
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
12A. Composition rules: Gauss, Dirichlet, and Bhargava
12.7. Composition and Gauss
12.8. Dirichlet composition
12.9. Bhargava composition
12B. The class group
12.10. A dictionary between binary quadratic forms and ideals
12.11. Elements of order two in the class group
References for this chapter
12C. Binary quadratic forms of positive discriminant
12.12. Binary quadratic forms with positive discriminant, and continued fractions
12.13. The set of automorphisms
12D. Sums of three squares
12.14. Connection between sums of 3 squares and ℎ(𝑑)
12.15. Dirichlet’s class number formula
12E. Sums of four squares
12.16. Sums of four squares
12.17. Quaternions
12.18. The number of representations
12F. Universality
12.19. Universality of quadratic forms
References for this appendix
12G. Integers represented in Apollonian circle packings
12.20. Combining these linear transformations
Further reading on Apollonian packings
Chapter 13. The anatomy of integers
13.1. Rough estimates for the number of integers with a fixed number of prime factors
13.2. The number of prime factors of a typical integer
13.3. The multiplication table problem
13.4. Hardy and Ramanujan’s inequality
13A. Other anatomies
13.5. The anatomy of polynomials in finite fields
13.6. The anatomy of permutations
More on mathematical anatomies
13B. Dirichlet 𝐿-functions
13.7. Dirichlet series
Further exercises
Chapter 14. Counting integral and rational points on curves, modulo 𝑝
14.1. Diagonal quadratics
14.2. Counting solutions to a quadratic equation and another proof of quadratic reciprocity
14.3. Cubic equations modulo 𝑝
14.4. The equation 𝐸_{𝑏}:𝑦²=𝑥³+𝑏
14.5. The equation 𝑦²=𝑥³+𝑎𝑥
14.6. A more general viewpoint on counting solutions modulo 𝑝
14A. Gauss sums
14.7. Identities for Gauss sums
14.8. Dirichlet 𝐿-functions at 𝑠=1
14.9. Jacobi sums
14.10. The diagonal cubic, revisited
Chapter 15. Combinatorial number theory
15.1. Partitions
15.2. Jacobi’s triple product identity
15.3. The Freiman-Ruzsa Theorem
15.4. Expansion and the Plünnecke-Ruzsa inequality
15.5. Schnirel′man’s Theorem
15.6. Classical additive number theory
15.7. Challenging problems
Further reading for chapter 15
15A. Summing sets modulo 𝑝
15.8. The Cauchy-Davenport Theorem
15B. Summing sets of integers
15.9. The Frobenius postage stamp problem, III
Chapter 16. The 𝑝-adic numbers
16.1. The 𝑝-adic norm
16.2. 𝑝-adic expansions
16.3. 𝑝-adic roots of polynomials
16.4. 𝑝-adic factors of a polynomial
Factoring polynomials in ℤ[𝕩] efficiently
Further reading on factoring polynomials
16.5. Possible norms on the rationals
16.6. Power series convergence and the 𝑝-adic logarithm
16.7. The 𝑝-adic dilogarithm
Further reading on 𝑝-adics
Chapter 17. Rational points on elliptic curves
17.1. The group of rational points on an elliptic curve
17.2. Congruent number curves
17.3. No non-trivial rational points by descent
17.4. The group of rational points of 𝑦²=𝑥³-𝑥
17.5. Mordell’s Theorem: 𝐸_{𝐴}(ℚ) is finitely generated
Much of the discussion in this chapter is developed from
17.6. Some nice examples
Further reading on the basics of elliptic curves
17A. General Mordell’s Theorem
17.7. The growth of points
17B. Pythagorean triangles of area 6
Integer points
There are many techniques to limit integer points in
17C. 2-parts of abelian groups
17.8. 2-parts of abelian, arithmetic groups
17D. Waring’s problem
17.9. Waring’s problem
Further reading on Waring’s problem
Hints for exercises
Recommended further reading
Index
Back Cover

I am a math teacher with a particular interest in number theory. I bought 'Number Theory Revealed: A Masterclass' just the other day, and it's already my favorite book on the subject. The amount of fascinating topics covered is really remarkable - it goes far beyond any other number theory book I've read, and yet it feels quite approachable.

Aaron Doman, Art of Problem Solving
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

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