Number Theory Structures, Examples, and Problems
Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is
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Theory Structures, Examples, and Problems
Birkhäuser Boston • Basel • Berlin
Dorin Andrica Faculty of Mathematics and Computer Science “Babeş-Bolyai” University Str. Kogalniceanu 1 3400 Cluj-Napoca, Romania [email protected]
Titu Andreescu Department of Science and Mathematics Education University of Texas at Dallas 2601 N. Floyd Road, FN 33 Richardson, TX 75002, USA [email protected]
ISBN: 978-0-8176-3245-8 DOI: 10.1007/b11856
e-ISBN: 978-0-8176-4645-5
Library of Congress Control Number: 2009921128 Mathematics Subject Classification (2000): 11A05, 11A07, 11A15, 11A25, 11A25, 11B37, 11B39, 11B65, 11D04, 11D09, 11D25 © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Cover Design by Joseph Sherman Printed on acid-free paper. www.birkhauser.com
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God made the integers, all else is the work of man. Leopold Kronecker
Contents Preface
xiii
Acknowledgments
xv
Notation
xvii
I Fundamentals 1
Divisibility 1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Greatest Common Divisor and Least Common Multiple 1.4 Odd and Even . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . 1.6 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . 1.7 Numerical Systems . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Representation of Integers in an Arbitrary Base . . . 1.7.2 Divisibility Criteria in the Decimal System . . . . .
1 . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
3 3 9 17 27 29 34 36 36 38
2
Powers of Integers 47 2.1 Perfect Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Perfect Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3 kth Powers of Integers, k at least 4 . . . . . . . . . . . . . . . . . 57
3
Floor Function and Fractional Part 61 3.1 General Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Floor Function and Integer Points . . . . . . . . . . . . . . . . . 68 3.3 A Useful Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
viii
Contents
4
Digits of Numbers 77 4.1 The Last Digits of a Number . . . . .
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