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19 Nineteen XIX aaa aaaa aaaaa aaaa aaa Corresponding ordinal: nineteenth. The number 19 is a centered hexagonal number as you can see. The number 19 is the tenth odd number, the eighth prime number, and the sixteenth deﬁcient number. The prime 19 is in the fourth pair of twin primes 17, 19. As the sum of four or fewer squares: 19 = 12 + 32 + 32 = 12 + 12 + 12 + 42. As the sum of nine or fewer cubes: 19 = 13 + 13 + 13 + 23 + 23. As the difference of two squares: 19 = 102 92. − The number 19 appears in only one Pythagorean triple, [19, 180, 181], which is primi- tive, of course. As a sum of three odd primes: 3 + 3 + 13, 3 + 5 + 11, and 5 + 5 + 7. The number 19 91 is equal to 1729, the smallest number that is the sum of two cubes × in two different ways. The number 19=1+4+9+4+1 is an octahedral number: The number 19 is the last of the “teen” numbers. The number 19 is the largest prime that is a palindrome in Roman numerals. The number 19 is the smallest number n such that nn is pandigital: 1919 = 1978 419 655 660 313 589 123 979. (Number Gossip) The number 19 is the smallest prime that is equal to the difference of two cubes of 2 Chapter 19 Nineteen XIX primes. (Number Gossip) It was established in 1986 that every number is the sum of at most 19 fourth powers. The number 79 requires 19 fourth powers: 79 = 15 14 + 4 24. · · The nineteenth President of the United States was Rutherford Birchard Hayes. The nineteenth state to enter the Union was Indiana. The nineteenth largest state in the United States is Washington. The nineteenth amendment to the U.S. Constitution says that “The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of sex.” The Hebrew calendar is based on a cycle of 19 years. The game of Go is played on a 19-by-19 board. The 19th card in the Major Arcana of the tarot deck is The Sun..

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~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used.
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