Number Theory Revealed: An Introduction

Andrew Granville

• Publisher: American Mathematical Society
• Year: 2019
• Print ISBN: 9781470441579
• eBook ISBN: 9781470454234
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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

• Cover
• Title page
• Preface
• Gauss’s Disquisitiones Arithmeticae
• Notation
• Prerequisites
• Preliminary Chapter on Induction
• Chapter 1. The Euclidean algorithm
• Appendix 1A. Reformulating the Euclidean algorithm
• Chapter 2. Congruences
• Appendix 2A. Congruences in the language of groups
• Chapter 3. The basic algebra of number theory
• Appendix 3A. Factoring binomial coefficients and Pascal’s triangle modulo 𝑝
• Chapter 4. Multiplicative functions
• Appendix 4A. More multiplicative functions
• Chapter 5. The distribution of prime numbers
• Appendix 5A. Bertrand’s postulate and beyond
• Bonus read: A review of prime problems
• Chapter 6. Diophantine problems
• Appendix 6A. Polynomial solutions of Diophantine equations
• Chapter 7. Power residues
• Appendix 7A. Card shuffling and Fermat’s Little Theorem
• Chapter 8. Quadratic residues
• Appendix 8A. Eisenstein’s proof of quadratic reciprocity
• Chapter 9. Quadratic equations
• Appendix 9A. Proof of the local-global principle for quadratic equations
• Chapter 10. Square roots and factoring
• Appendix 10A. Pseudoprime tests using square roots of 1
• Chapter 11. Rational approximations to real numbers
• Appendix 11A. Uniform distribution
• Chapter 12. Binary quadratic forms
• Appendix 12A. Composition rules: Gauss, Dirichlet, and Bhargava
• Hints for exercises
• Recommended further reading
• Index
• Back Cover