Number Theory Revealed An Introduction 1st edition by Andrew 1470454238 9781470454234

Number Theory Revealed An Introduction 1st edition by Andrew Granville – Ebook Instant Download/Delivery ISBN(s): 1470454238, 9781470454234

Product details:

• ISBN 10: 1470454238
• ISBN 13: 9781470454234
• Author: Andrew Granville

Number Theory Revealed: An Introduction acquaints undergraduates with the “Queen of Mathematics”. The text offers 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 $p$ and modern twists on traditional questions like the values represented by binary quadratic forms and large solutions of equations. Each chapter includes an “elective appendix” with additional reading, projects, and references. An expanded edition, Number Theory Revealed: A Masterclass, offers a more comprehensive approach to these core topics and adds additional material in further chapters and appendices, allowing instructors to create an individualized course tailored to their own (and their students’) interests. 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.

Table of contents:

Chapter 1. The Euclidean algorithm

1.1. Finding the gcd

1.2. Linear combinations

1.3. The set of linear combinations of two integers

1.4. The least common multiple

1.5. Continued fractions

1.6. Tiling a rectangle with squares

Additional exercises

Divisors in recurrence sequences

Appendix 1A. Reformulating the Euclidean algorithm

1.8. Euclid matrices and Euclid’s algorithm

1.9. Euclid matrices and ideal transformations

1.10. The dynamics of the Euclidean algorithm

Chapter 2. Congruences

2.1. Basic congruences

2.2. The trouble with division

2.3. Congruences for polynomials

2.4. Tests for divisibility

Additional exercises

Binomial coefficients modulo 𝑝

The Fibonacci numbers modulo 𝑑

Appendix 2A. Congruences in the language of groups

2.6. Further discussion of the basic notion of congruence

2.7. Cosets of an additive group

2.8. A new family of rings and fields

2.9. The order of an element

Chapter 3. The basic algebra of number theory

3.1. The Fundamental Theorem of Arithmetic

3.2. Abstractions

3.3. Divisors using factorizations

3.4. Irrationality

3.5. Dividing in congruences

3.6. Linear equations in two unknowns

3.7. Congruences to several moduli

3.8. Square roots of 1 (mod 𝑛)

Additional exercises

Reference on the many proofs that √2 is irrational

Appendix 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝

3.10. The prime powers dividing a given binomial coefficient

3.11. Pascal’s triangle modulo 2

References for this chapter

Chapter 4. Multiplicative functions

4.1. Euler’s 𝜙-function

4.2. Perfect numbers. “The whole is equal to the sum of its parts.”

Additional exercises

Appendix 4A. More multiplicative functions

4.4. Summing multiplicative functions

4.5. Inclusion-exclusion and the Möbius function

4.6. Convolutions and the Möbius inversion formula

4.7. The Liouville function

Additional exercises

Chapter 5. The distribution of prime numbers

5.1. Proofs that there are infinitely many primes

5.2. Distinguishing primes

5.3. Primes in certain arithmetic progressions

5.4. How many primes are there up to 𝑥?

5.5. Bounds on the number of primes

5.6. Gaps between primes

Further reading on hot topics in this section

5.7. Formulas for primes

Additional exercises

Appendix 5A. Bertrand’s postulate and beyond

5.9. Bertrand’s postulate

5.10. The theorem of Sylvester and Schur

Bonus read: A review of prime problems

5.11. Prime problems

Prime values of polynomials in one variable

Prime values of polynomials in several variables

Goldbach’s conjecture and variants

Other questions

Guides to conjectures and the Green-Tao Theorem

Chapter 6. Diophantine problems

6.1. The Pythagorean equation

6.2. No solutions to a Diophantine equation through descent

No solutions through prime divisibility

No solutions through geometric descent

6.3. Fermat’s “infinite descent”

6.4. Fermat’s Last Theorem

A brief history of equation solving

References for this chapter

Additional exercises

Appendix 6A. Polynomial solutions of Diophantine equations

6.6. Fermat’s Last Theorem in ℂ[𝕥]

6.7. 𝑎+𝑏=𝑐 in ℂ[𝕥]

Chapter 7. Power residues

7.1. Generating the multiplicative group of residues

7.2. Fermat’s Little Theorem

7.3. Special primes and orders

7.4. Further observations

7.5. The number of elements of a given order, and primitive roots

7.6. Testing for composites, pseudoprimes, and Carmichael numbers

7.7. Divisibility tests, again

7.8. The decimal expansion of fractions

7.9. Primes in arithmetic progressions, revisited

References for this chapter

Additional exercises

Appendix 7A. Card shuffling and Fermat’s Little Theorem

7.11. Card shuffling and orders modulo 𝑛

7.12. The “necklace proof” of Fermat’s Little Theorem

More combinatorics and number theory

7.13. Taking powers efficiently

7.14. Running time: The desirability of polynomial time algorithms

Chapter 8. Quadratic residues

8.1. Squares modulo prime 𝑝

8.2. The quadratic character of a residue

8.3. The residue -1

8.4. The residue 2

8.5. The law of quadratic reciprocity

8.6. Proof of the law of quadratic reciprocity

8.7. The Jacobi symbol

8.8. The squares modulo 𝑚

Additional exercises

Further reading on Euclidean proofs

Appendix 8A. Eisenstein’s proof of quadratic reciprocity

8.10. Eisenstein’s elegant proof, 1844

Chapter 9. Quadratic equations

9.1. Sums of two squares

9.2. The values of 𝑥²+𝑑𝑦²

9.3. Is there a solution to a given quadratic equation?

9.4. Representation of integers by 𝑎𝑥²+𝑏𝑦² with 𝑥,𝑦 rational, and beyond

9.5. The failure of the local-global principle for quadratic equations in integers

9.6. Primes represented by 𝑥²+5𝑦²

Additional exercises

Appendix 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

Chapter 10. Square roots and factoring

10.1. Square roots modulo 𝑛

10.2. Cryptosystems

10.3. RSA

10.4. Certificates and the complexity classes P and NP

10.5. Polynomial time primality testing

10.6. Factoring methods

References: See [CP05] and [Knu98], as well as:

Additional exercises

Appendix 10A. Pseudoprime tests using square roots of 1

10.8. The difficulty of finding all square roots of 1

Chapter 11. Rational approximations to real numbers

11.1. The pigeonhole principle

11.2. Pell’s equation

11.3. Descent on solutions of 𝑥²-𝑑𝑦²=𝑛,𝑑>0

11.4. Transcendental numbers

11.5. The 𝑎𝑏𝑐-conjecture

Further reading for this chapter

Additional exercises

Appendix 11A. Uniform distribution

11.7. 𝑛𝛼 mod 1

11.8. Bouncing billiard balls

Chapter 12. Binary quadratic forms

12.1. Representation of integers by binary quadratic forms

12.2. Equivalence classes of binary quadratic forms

12.3. Congruence restrictions on the values of a binary quadratic form

12.4. Class numbers

12.5. Class number one

References for this chapter

Additional exercises

Appendix 12A. Composition rules: Gauss, Dirichlet, and Bhargava

12.7. Composition and Gauss

12.8. Dirichlet composition

12.9. Bhargava composition

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