Problem-Solving and Selected Topics in Number Theory In the Spirit o
This book is designed to introduce some of the most important theorems and results from number theory while testing the reader’s understanding through carefully selected Olympiad-caliber problems. These problems and their solutions provide the reader with
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Michael Th. Rassias
Problem-Solving and Selected Topics in Number Theory In the Spirit of the Mathematical Olympiads
( Foreword by Preda Mihailescu
Michael Th. Rassias Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB, UK [email protected]
ISBN 978-1-4419-0494-2 e-ISBN 978-1-4419-0495-9 DOI 10.1007/978-1-4419-0495-9 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): 11-XX, 00A07 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (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.
Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my father Themistocles
Contents
Foreword by Preda Mih˘ ailescu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic methods to compute the greatest common divisor . . . . . . 1.2.1 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Blankinship’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . 1.4 Rational and irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 4 5 5 6 8
2
Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The M¨ obius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Euler function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The τ -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The generalized σ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 15 16 20 24 26
3
Perfect numbers, Fermat numbers . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Perfect numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Related ope
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