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Elementary Number Theory PDF

407 Pages·2015·3.665 MB·English
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Mathematics TEXTBOOKS in MATHEMATICS TEXTBOOKS in MATHEMATICS Elementary Number Theory takes an accessible approach to teaching E readers about the role of number theory in pure mathematics and its im- l Elementary e portant applications to cryptography and other areas. The first chapter m of the book explains how to do proofs and includes a brief discussion e of lemmas, propositions, theorems, and corollaries. The core of the text Number n covers linear Diophantine equations; unique factorization; congruences; t Fermat’s, Euler’s, and Wilson’s theorems; order and primitive roots; and a quadratic reciprocity. The authors also discuss numerous cryptographic Theory r topics, such as RSA and discrete logarithms, along with recent develop- y ments. N • Describes exciting applications of number theory to various problems in cryptography u • Discusses recent developments m • Includes “check your understanding” problems that test readers on b fundamental ideas, with solutions at the end of each chapter e • Provides numerous examples as well as standard and r computational exercises, with selected answers or hints in an T appendix h • Reviews the prerequisites in an appendix e About the Authors o James S. Kraft teaches mathematics at the Gilman School. He has pre- r viously taught at the University of Rochester, St. Mary’s College of Cali- y fornia, and Ithaca College. He has also worked in communications secu- rity. Dr. Kraft has published several research papers in algebraic number theory. He earned his Ph.D. from the University of Maryland. W Lawrence C. Washington is a professor of mathematics and Distin- a guished Scholar-Teacher at the University of Maryland. Dr. Washington sK h has published extensively in number theory, including books on cryp- ir na James S. Kraft tography (with Wade Trappe), cyclotomic fields, and elliptic curves. He gf t t earned his Ph.D. from Princeton University. o Lawrence C. Washington n K24624 www.crcpress.com TEXTBOOKS in MATHEMATICS Elementary Number Theory James S. Kraft Gilman School Baltimore, Maryland, USA Lawrence C. Washington University of Maryland College Park, Maryland, USA K24624_FM.indd 1 10/22/14 12:45 PM TEXTBOOKS in MATHEMATICS Series Editors: Al Boggess and Ken Rosen PUBLISHED TITLES ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH William Paulsen ADVANCED CALCULUS: THEORY AND PRACTICE John Srdjan Petrovic ADVANCED LINEAR ALGEBRA Nicholas Loehr ANALYSIS WITH ULTRASMALL NUMBERS Karel Hrbacek, Olivier Lessmann, and Richard O’Donovan APPLYING ANALYTICS: A PRACTICAL APPROACH Evan S. Levine COMPUTATIONS OF IMPROPER REIMANN INTEGRALS Ioannis Roussos CONVEX ANALYSIS Steven G. Krantz COUNTEREXAMPLES: FROM ELEMENTARY CALCULUS TO THE BEGINNINGS OF ANALYSIS Andrei Bourchtein and Ludmila Bourchtein DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION Steven G. Krantz DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster ELEMENTARY NUMBER THEORY James S. Kraft and Lawrence C. Washington ELEMENTS OF ADVANCED MATHEMATICS, THIRD EDITION Steven G. Krantz EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala K24624_FM.indd 2 10/22/14 12:45 PM PUBLISHED TITLES CONTINUED AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY James Kraft and Larry Washington AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB®, SECOND EDITION Mathew Coleman INTRODUCTION TO THE CALCULUS OF VARIATIONS AND CONTROL WITH MODERN APPLICATIONS John T. Burns LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION Bruce Solomon THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY David G. Taylor QUADRACTIC IRRATIONALS: AN INTRODUCTION TO CLASSICAL NUMBER THEORY Franz Holter-Koch REAL ANALYSIS AND FOUNDATIONS, THIRD EDITION Steven G. Krantz RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION Bilal M. Ayyub RISK MANAGEMENT AND SIMULATION Aparna Gupta TRANSFORMATIONAL PLANE GEOMETRY Ronald N. Umble and Zhigang Han K24624_FM.indd 3 10/22/14 12:45 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20141007 International Standard Book Number-13: 978-1-4987-0269-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Kristi, Danny, and Aaron, and to Miriam Kraft and the memory of Norman Kraft To Susan and Patrick, and to Ida H. Washington and Lawrence M. Washington Contents Preface xiii Introduction 1 1 Divisibility 9 1.1 What Is a Proof? . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Proof by Contradiction . . . . . . . . . . . . . . 15 1.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Euclid’s Theorem . . . . . . . . . . . . . . . . . . . . . 19 1.4 Euclid’s Original Proof . . . . . . . . . . . . . . . . . . 21 1.5 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . 23 1.6 The Division Algorithm . . . . . . . . . . . . . . . . . . 25 1.6.1 A Cryptographic Application . . . . . . . . . . . 26 1.7 The Greatest Common Divisor . . . . . . . . . . . . . . 27 1.8 The Euclidean Algorithm . . . . . . . . . . . . . . . . . 30 1.8.1 The Extended Euclidean Algorithm . . . . . . . 32 1.9 Other Bases . . . . . . . . . . . . . . . . . . . . . . . . 37 1.10 Fermat and Mersenne Numbers . . . . . . . . . . . . . 40 1.11 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 43 1.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.12.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 44 1.12.2 Computer Explorations . . . . . . . . . . . . . . 50 1.12.3 Answers to “Check Your Understanding” . . . . 51 2 Linear Diophantine Equations 55 2.1 ax+by = c . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 The Postage Stamp Problem . . . . . . . . . . . . . . . 60 vii viii Contents 2.3 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 64 2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 64 2.4.2 Answers to “Check Your Understanding” . . . . 66 3 Unique Factorization 67 3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . 67 3.2 The Fundamental Theorem of Arithmetic . . . . . . . . 69 3.3 Euclid and the Fundamental Theorem of Arithmetic . . 74 3.4 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 75 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 75 3.5.2 Answers to “Check Your Understanding” . . . . 76 4 Applications of Unique Factorization 77 4.1 A Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Irrationality Proofs . . . . . . . . . . . . . . . . . . . . 79 4.3 The Rational Root Theorem . . . . . . . . . . . . . . . 81 4.4 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . 83 4.5 Differences of Squares . . . . . . . . . . . . . . . . . . . 87 4.6 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 89 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 90 4.7.2 Computer Explorations . . . . . . . . . . . . . . 93 4.7.3 Answers to “Check Your Understanding” . . . . 93 5 Congruences 95 5.1 Definitions and Examples . . . . . . . . . . . . . . . . . 95 5.2 Modular Exponentiation . . . . . . . . . . . . . . . . . 102 5.3 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Linear Congruences . . . . . . . . . . . . . . . . . . . . 108 5.5 The Chinese Remainder Theorem . . . . . . . . . . . . 115 5.6 Fractions Mod m . . . . . . . . . . . . . . . . . . . . . 119

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