Concrete mathematics

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Dedicated to Leonhard Euler (1707-1 783)


Dedicated to Leonhard Euler (1707-1 783)


Ronald L. Graham

AT&T Bell Laboratories

Donald E. Knuth

Stanford University

Oren Patashnik

Stanford University ▲


ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan

Library of Congress Cataloging-in-Publication Data Graham, Ronald Lewis, 1935-

Concrete mathematics : a foundation for computer science / Ronald L. Graham, Donald E. Knuth, Oren Patashnik. xiii,625 p. 24 cm.

Bibliography: p. 578 Includes index.

ISBN O-201-14236-8

III. Title.


1. Mathematics—1961- 2. Electronic data processing—Mathematics. I. Knuth, Donald Ervin, 1938-    II. Patashnik, Oren, 1954-



Sixth printing, with corrections, October 1990 Copyright © 1989 by Addison-Wesley Publishing Company

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Published simultaneously in Canada.


“A odience, level, and treatment — a description of such matters is what prefaces are supposed to be about.”

— P. R. Halmos [142]

“People do acquire a little brief authority by equipping

themselves with jargon: they can pontificate and air я superficial expertise.

But what we should ask of educated mathematicians is not what they can speechify about, nor even what they know about the existing corpus of mathematical knowledge, but rather what can they now do with their learning and whether they can actually solve mathematical problems arising in practice.

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