# From zero to infinity what makes

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What mathematicians and teachers write about

FROM ZERO TO INFINITY

I read From Zero to Infinity when I was a schoolboy in Oxford,England, and my only regret is that I was well into my teens (17)before it happened. Just last week I gave away my most recentcopy to the 12-year-old daughter of a friend. I will be getting an-other copy for myself as soon as I can.

John B. Cosgrave, St. Patricks College, Dublin, Ireland

After reading From Zero to Infinity, I was hooked. This book dis-cussed many beautiful ideas and facts about the integers and posedseveral interesting problems. I tried to solve them. I failed. I triedto construct counter examples. I made my own conjectures andproved some related results. By the time I graduated from highschool I had filled two 2-inch notebooks with my own ideas, re-sults, and calculations.

Nathaniel Dean, Mathematics Department,Bell Communications Research

Constance Reids book From Zero to Infinity was translated intoJapanese, and I found it when I was a junior high school boy. Iwas really impressed by Reids book, and I read it repeatedly. In-spired by it, I even tried to solve Fermats Last Theorem. Now Iam working in analytic number theory, and I think one of the rea-sons for my choice is a sentence in Reids book: Analytic numbertheory is said to be technically the most difficult in the whole ofmathematics.

Kohji Matsumoto, student at Nagoya University

Yesterday, aboard a flight from Denver, I had a nice conversationwith a gentleman, John Moulter. When I learned that Mr. Moulteris a retired Los Angeles mathematics teacher, I mentioned that Ihad the good fortune to know Connie Reid. Mr. Moulter abruptlydemanded, Do you mean Constance R-E-I-D? When I answeredaffirmatively, he told me that you had changed his life. Aboutfifty years ago, then a high-school history teacher, he picked up acopy of Esquire magazine in a barbershop containing a review ofFrom Zero to Infinity. The review prompted him to buy your book.And your book inspired him to switch from teaching history toteaching mathematics!

Letter from a friend with appended notefrom John Moulter: Thank you. Thank you.

I was the sort of child who always carried a book wherever hewent. In fifth and sixth grade that book was most frequently FromZero to Infinity. I was indeed born to be a mathematician, but FromZero to Infinity helped me to realize that I was part of a communityof number-people. There can be few pleasures more satisfyingthan having the chance, as an adult, to help bring ones favoritechildhood book back into print.

Bruce Reznick, University of Illinois Urbana

I want to thank you for having written such a wonderful book. Itwas pitched on just the right level for a young teenager but, moreto the point, it expressed the right mix of beauty and wonder. I justhad to learn more. I very much believe that this small book, whichstill occupies an important place in my personal library, enrichedmy life immeasurably. It is very rare that we find what we reallywant to do in life, and I am very grateful that your book led me inthe right direction.

Hugh Williams, University of Calgary, Canada,author of Edouard Lucas and Primality Testing

F R O M Z E R O T O I N F I N I T Y

F I F T I E T H A N N I V E R S A R Y E D I T I O N

From Zero to InfinityWhat Makes Numbers Interesting

Constance Reid

A K Peters, Ltd.Wellesley, Massachusetts

Editorial, Sales, and Customer Service Office

A K Peters, Ltd.888 Worcester Street, Suite 230Wellesley, MA 02482www.akpeters.com

Copyright c2006 by A K Peters, Ltd.

All rights reserved. No part of the material protected by this copyrightnotice may be reproduced or utilized in any form, electronic or mechani-cal, including photocopying, recording, or by any information storage andretrieval system, without written permission from the copyright owner.

First published in 1955 by Thomas Y. Crowell Co.Second edition, 1960 by Thomas Y. Crowell Co.Third edition, 1964 by Thomas Y. Crowell Co.Fourth edition, 1992 by The Mathematical Association of America.Fifth edition, 2006

Library of Congress Cataloging-in-Publication Data

Reid, Constance.From zero to infinity : what makes numbers interesting /

Constance Reid. 5th ed.p. cm.

ISBN-13: 978-1-56881-273-1 (alk. paper)ISBN-10: 1-56881-273-6 (alk. paper)1. Numerals. 2. Number theory. I. Title.QA93.R42 2006.510--dc22 2005027860

Printed in the United States of America10 09 08 07 06 10 9 8 7 6 5 4 3 2 1

I N M E M O R Y O F

Julia Bowman Robinson19191985

and

Raphael Mitchel Robinson19111995

C O N T E N T S

A C K N O W L E D G M E N T S xi

A U T H O R S N O T E xiii

0 Z E R O 1Only a place-holder until finally recognizedas first of the natural numbers.

1 O N E 15A number that makes mathematics differentfrom all the other sciences.

2 T W O 27A primitive number system comes into its ownwith the electronic computer.

3 T H R E E 39First among the odd numbers divisible only bythemselves and one.

4 F O U R 53Numbers multiplied by themselves providebeautifully difficult theorems.

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F R O M Z E R O T O I N F I N I T Y

5 F I V E 67The pentagonal numbers turn up in thegenerating function of partitions.

6 S I X 77Why has finding larger perfect numbersbecome increasingly important?

7 S E V E N 91The problem of the prime-sided regular polygonshas an unexpected answer.

8 E I G H T 105Solving one problem about cubes leads to aneven more difficult problem.

9 N I N E 117Add a third line to the two lines of theequals signand see what happens.

e E U L E R S N U M B E R 131An unnatural number answers the deepestquestion about natural numbers.

0 A L E P H - Z E R O 149What set greater than the positive integershas the same number?

I N D E X 169

viii

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

The natural numbers, which are the primary subject of thisbook, do not end with the digits with which we representthem. They continue indefinitelyas the three dots indi-cateto infinity. And they are all interesting: for if therewere any uninteresting numbers, there would of necessitybe a smallest uninteresting number and it, for that reasonalone, would be very interesting.

A C K N O W L E D G M E N T S

Throughout the half century during which From Zero to In-finity has been in print, it has had a number of differentpublishers. As its author I would like to express my spe-cial gratitude to four of them.

First, to Dennis Flanagan, the longtime editor of Scien-tific American, who accepted an article on Perfect Num-bers from a freelancing housewife who was not even amathematician.

Second, to Robert L. Crowell, who read her article in Sci-entific American, saw its possibilities for a book, and shep-herded it and its author through three editions.

Third, to Donald J. Albers, publications director of theMathematical Association of America, who, after I had re-trieved the copyright from the last in the series of publish-ing companies that had come into possession of Crowellbooks, published a fourth edition of From Zero to Infinityunder the imprint of the MAA.

Fourth, to Klaus Peters, the president of A K Peters, Ltd.I am particularly happy that Klaus will be republishing

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F R O M Z E R O T O I N F I N I T Y

my first mathematical book, because he was also the pub-lisher, as mathematics editor of Springer-Verlag, who in1969 accepted for publication my life of David Hilbert andthus opened up to me a new field of mathematical writingthe writing of mathematical lives.

Constance Reid

xii

A U T H O R S N O T E

There is a story behind the publication of this fiftiethanniversary edition of From Zero to Infinity.

It begins with a phone call from my sister, Julia Robin-son, on the morning of January 31, 1952. She has called totell me of an exciting event that occurred the night beforeat the Institute for Numerical Analysis on the UCLA cam-pus, where the National Bureau of Standards has located itsWestern Automatic Computerthe SWAC.

Julia tells me that a program by her husband, RaphaelRobinson, had turned up the first new perfect numbersin seventy-five yearsnot one but two of them. (I learnonly later from others that Raphael had at this point neverseen the SWAC and had programmed entirely from a copyof the manual.) Julia explains the problem simply: perfectnumbersthe name itself is intriguingare numbers like6 that are the sum of all their divisors except themselves:6 = 1 + 2 + 3. Then she tells me there is a particular formof prime necessary for the formation of such numbers, theamount of calculation involved in determining their pri-mality, the enormousness of such primes. For me the whole

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F R O M Z E R O T O I N F I N I T Y

thing is fascinating. I decide to write an article about thediscovery of new perfect numbers.

I am lucky to be able to interview Dick Lehmer, theDirector of the SWAC, while he and his wife Emma are vis-iting in Berkeley. It is Emma who suggests that I send myarticle to Scientific American. If you look up the March 1953issue you will see a photo of the SWAC and be able to reada fairly detailed description of Raphaels program and howit was fed into the computer.

Of course a subscriber later wrote to Dennis Flanagan,the editor, to complain that when he read an article in Sci-entific American he expected the author to be a Ph.D. But mynot being a Ph.D. did not seem to have concerned DennisFlanagan anymore than it concerned the publisher RobertL. Crowell. After reading my article, Mr. Crowell immedi-ately wrote to ask if I would be inte