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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
Number Theory Revealed: A Masterclass
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
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
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
Number Theory Revealed: A Masterclass
Click above image for expanded view
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
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
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
  • Book Details
     
     
    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.

  • 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
    • 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
  • Reviews
     
     
    • 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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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
  • 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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