Avner Ash - Summing It Up: From One Plus One to Modern Number Theory

SUMMING IT UP

SUMMING IT UP

From One Plus One to Modern Number Theory

AVNER ASH AND ROBERT GROSS

PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD

Copyright 2016 by Princeton University Press

Published by Princeton University Press, 41 William Street,

Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press, 6 Oxford Street,
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Library of Congress Cataloging-in-Publication Data

Names: Ash, Avner, 1949 | Gross, Robert, 1959

Title: Summing it up : from one plus one to modern number theory / Avner Ash and Robert Gross.

Description: Princeton : Princeton University Press, [2016] | Includes bibliographical references and index.

Identifiers: LCCN 2015037578 | ISBN 9780691170190 (hardcover)

Subjects: LCSH: Number theory. | MathematicsPopular works.

Classification: LCC QA241 .A85 2016 | DDC 512.7dc23 LC record available at http://lccn.loc.gov/2015037578

British Library Cataloging-in-Publication Data is available

This book has been composed in New Century Schoolbook

Printed on acid-free paper.

Typeset by S R Nova Pvt Ltd, Bangalore, India

Printed in the United States of America

1 3 5 7 9 10 8 6 4 2

For our families

Manners are not taught in lessons, said Alice. Lessons teach you to do sums, and things of that sort.

And you do Addition? the White Queen asked. Whats one and one and one and one and one and one and one and one and one and one?

I dont know, said Alice. I lost count.

She cant do Addition, the Red Queen interrupted.

Lewis Carroll, Through the Looking-Glass

Numbers it is. All music when you come to think. Two multiplied by two divided by half is twice one. Vibrations: chords those are.

One plus two plus six is seven. Do anything you like with figures juggling. Always find out this equal to that. Symmetry under a cemetery wall.

James Joyce, Ulysses

my true love is grown to such excess

I cannot sum up sum of half my wealth.

William Shakespeare, Romeo and Juliet, II.vi.3334

Contents

Preface

Adding two whole numbers together is one of the first things we learn in mathematics. Addition is a rather simple thing to do, but it almost immediately raises all kinds of curious questions in the mind of an inquisitive person who has an inclination for numbers. Some of these questions are listed in the Introduction on page 3. The first purpose of this book is to explore in a leisurely way these and related questions and the theorems they give rise to. You may read a more detailed discussion of our subject matter in the Introduction.

It is in the nature of mathematics to be precise; a lack of precision can lead to confusion. For example, in Blooms musings from Ulysses, quoted as one of our epigraphs, a failure to use exact language and provide full context makes it appear that his arithmetical claims are nonsense. They may be explained, however, as follows: Two multiplied by two divided by half is twice one means , because Bloom meant cut in half when he used the ambiguous phrase divided by half. Then 1 + 2 + 6 = 7 refers to musical intervals: If you add a unison, a second, and a sixth, the result is indeed a seventh. Bloom himself notes that this can appear to be juggling if you suppress a clear indication of what you are doing.

Bloom has the luxury of talking to himself. In this book, we strive to be clear and precise without being overly pedantic. The reader will decide to what extent we have succeeded. In addition to clarity and precision, rigorously logical proofs are characteristic of mathematics. All of the mathematical assertions in this book can be proved, but the proofs often are too intricate for us to discuss in any detail. In a textbook or research monograph, all such proofs would be given, or reference made to places where they could be found. In a book such as this, the reader must trust us that all of our mathematical assertions have proofs that have been verified.

This book is the third in a series of books about number theory written for a general mathematically literate audience. (We address later exactly what we mean by mathematically literate.) The first two books were Fearless Symmetry and Elliptic Tales (Ash and Gross, 2006; 2012). The first book discussed problems in Diophantine equations, such as Fermats Last Theorem (FLT). The second discussed problems related to elliptic curves, such as the BirchSwinnerton-Dyer Conjecture. In both of these books, we ended up mumbling something about modular forms, an advanced topic that plays a crucial role in both of these areas of number theory. By the time we reached the last chapters in these books, we had already introduced so many concepts that we could only allude to the theory of modular forms. One purpose of Summing It Up is to give in .

Each of the three books in our trilogy may be read independently of the others. After reading the first two parts of Summing It Up, a very diligent person might gain from reading Fearless Symmetry or Elliptic Tales in tandem with the third part of Summing It Up, for they provide additional motivation for learning about modular forms, which are dealt with at length in . Of course, this is not necessarywe believe that the number-theoretical problems studied in the first two parts by themselves lead naturally to a well-motivated study of modular forms.

The three parts of this book are designed for readers of varying degrees of mathematical background. does not require any additional mathematical knowledge, but it gets rather intricate. You may need a good dose of patience to read through all of the details.

The level of difficulty of the various chapters and sections sometimes fluctuates considerably. You are invited to browse them in any order. You can always refer back to a chapter or section you skipped, if necessary, to fill in the details. However, in , things will probably make the most sense if you read the chapters in order.

It continues to amaze us what human beings have accomplished, starting with one plus one equals two, getting to two plus two equals four (the clich example of a simple truth that we know for sure is true), and going far beyond into realms of number theory that even now are active areas of research. We hope you will enjoy our attempts to display some of these wonderful ideas in the pages that follow.

Acknowledgments

We wish to thank the anonymous readers employed by Princeton University Press, all of whom gave us very helpful suggestions for improving the text of this book. Thanks to Ken Ono and David Rohrlich for mathematical help, Richard Velkley for philosophical help, and Betsy Blumenthal for editorial help. Thanks to Carmina Alvarez and Karen Carter for designing and producing our book. Great thanks, as always, to our editor, Vickie Kearn, for her unfailing encouragement.

SUMMING IT UP

Introduction

WHAT THIS BOOK IS ABOUT

1. Plus

Countingone, two, three, four or uno, dos, tres, cuatro (or in whatever language); or I, II, III, IV or 1, 2, 3, 4, or in whatever symbolsis probably the first theoretical mathematical activity of human beings. It is theoretical because it is detached from the objects, whatever they might be, that are being counted. The shepherd who first piled up pebbles, one for each sheep let out to graze, and then tossed them one by one as the sheep came back to the fold, was performing a practical mathematical actcreating a one-to-one correspondence. But this act was merely practical, without any theory to go with it.