Number theory revealed: a masterclass

617 Pages·2019·4.687 MB·English

by  Granville A

Number Theory Revealed:
A Masterclass
Andrew Granville
Number Theory Revealed:
A Masterclass

Number Theory Revealed:
A Masterclass
Andrew Granville
Cover design by Marci Babineau.
Front cover image of Srinivasa Ramanujan in the playing card: Oberwolfach Photo
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Front cover image of Andrew Wiles in playing card, credit: AlainGoriely.
2010 Mathematics Subject Classification. Primary 11-01, 11A55, 11B30,11B39, 11D09,
11D25, 11N05, 11N25.
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Dedicated to my beloved wife, Marci.
Writing this book has had its challenges.
Being the spouse of the author, while writing
this book, has also had its challenges.

The enchanting charms of this sublime science
reveal themselves only to those who have the
courage to go deeply into it.
Carl Friedrich Gauss, 1807

Contents
Preface xvii
Gauss’s Disquisitiones Arithmeticae xxiii
Notation xxv
The language of mathematics xxvi
Prerequisites xxvii
Preliminary Chapter on Induction 1
0.1. Fibonacci numbers and other recurrence sequences 1
0.2. Formulas for sums of powers of integers 3
0.3. The binomial theorem, Pascal’s triangle, and the binomial coefficients 4
Appendices for Preliminary Chapter on Induction
0A. A closed formula for sums of powers 9
0B. Generating functions 11
0C. Finding roots of polynomials 15
0D. What is a group? 19
0E. Rings and fields 22
0F. Symmetric polynomials 25
0G. Constructibility 30
Chapter 1. The Euclidean algorithm 33
1.1. Finding the gcd 33
1.2. Linear combinations 35
1.3. The set of linear combinations of two integers 37
1.4. The least common multiple 39
ix