Elementary Number Theory andlts Applications KennethH. Rosen AT&T Informotion SystemsLaboratories (formerly part of Bell Laborotories) A YY ADDISON-WESLEY PUBLISHING COMPANY Read ing, Massachusetts Menlo Park, California London Amsterdam Don Mills, Ontario Sydney Cover: The iteration of the transformation n/2 if n T(n) : \ is even l Qn + l)/2 if n is odd is depicted.The Collatz conjectureasserts that with any starting point, the iteration of ?"eventuallyreaches the integer one. (SeeProblem 33 of Section l.2of the text.) Library of Congress Cataloging in Publication Data Rosen, Kenneth H. Elementary number theory and its applications. Bibliography: p. Includes index. l. Numbers, Theory of. I. Title. QA24l.R67 1984 512',.72 83-l1804 rsBN 0-201-06561-4 Reprinted with corrections, June | 986 Copyright O 1984 by Bell Telephone Laboratories and Kenneth H. Rosen. 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 prior written permission of the publisher. printed in the United States of America. Published simultaneously in Canada. DEFGHIJ_MA_8987 Preface Number theory has long been a favorite subject for studentsand teachersof mathematics. It is a classical subject and has a reputation for being the "purest" part of mathematics, yet recent developments in cryptology and computer science are based on elementary number theory. This book is the first text to integrate these important applications of elementary number theory with the traditional topics covered in an introductory number theory course. This book is suitable as a text in an undergraduatenumber theory courseat any level. There are no formal prerequisitesneeded for most of the material covered, so that even a bright high-school student could use this book. Also, this book is designedto be a useful supplementarybook for computer science courses,and as a number theory primer for computer scientistsinterested in learning about the new developmentsin cryptography. Some of the important topics that will interest both mathematics and computer sciencestudents are recursion,algorithms and their computationai complexity, computer arithmetic with large integers, binary and hexadecimal representations of integers, primality testing, pseudoprimality,pseudo-random numbers, hashing functions, and cryptology, including the recently-invented area of public-key cryptography. Throughout the book various algorithms and their computational complexitiesare discussed.A wide variety of primality tests are developedin the text. Use of the Book The core material for a course in number theory is presentedin Chapters 1, 2, and 5, and in Sections3.1-3.3 and 6.1. Section 3.4 contains some linear algebra; this section is necessary background for Section 7.2; these two sections can be omitted if desired. Sections 4.1, 4.2, and 4.3 present traditional applications of number theory and Section 4.4 presents an application to computer science; the instructor can decide which of these sectionsto cover. Sections 6.2 and 6.3 discussarithmetic functions. Mersenne primes, and perfect numbers; some of this material is used in Chapter 8. Chapter 7 covers the applications of number theory to cryptology. Sections 7.1, 7.3, and 7.4, which contain discussionsof classical and public-key vt Preface cryptography,should be includedin all courses.Chapter 8 dealswith primitive roots; Sections 8.1-8.4 should be covered if possible. Most instructors will want to include Section 8.7 which deals with pseudo-randomnumbers. Sections 9.1 and 9.2 are about quadratic residues and reciprocity, a fundamental topic which should be covered if possible;Sections 9.3 and 9.4 deal with Jacobi symbols and Euler pseudoprimesand should interest most readers. Section 10.1, which coversrational numbers and decimal fractions. and Sections I 1.1 and I 1.2 which discussPythagorean triples and Fermat's last theorem are coveredin most number theory courses. Sections 10.2-10.4 and I 1.3 involve continued fractions; thesesections are optional. The Contents The reader can determine which chapters to study based on the following descriptionof their contents. Chapter I introduces two importants tools in establishingresults about the integers, the well-ordering property and the principle of mathematical induction. Recursivedefinitions and the binomial theorem are also developed. The concept of divisibility of integers is introduced. Representations of integers to different bases are described, as are algorithms for arithmetic operations with integers and their computational complexity (using big-O notation). Finally, prime numbers, their distribution, and conjectures about primes are discussed. Chapter 2 introduces the greatest common divisor of a set of integers. The Euclidean algorithm, used to find greatest common divisors, and its computational complexity, are discussed,as are algorithms to express the greatest common divisor as a linear combination of the integers involved. The Fibonacci numbers are introduced. Prime-factorizations, the fundamental theorem of arithmetic, and factorization techniques are covered. Finally, linear diophantine equationsare discussed. Chapter 3 introduces congruences and develops their fundamental properties. Linear congruencesin one unknown are discussed,as are systems of linear congruences in one or more unknown. The Chinese remainder theorem is developed,and its application to computer arithmetic with large integers is described. Chapter 4 developsapplications of.congruences. In particular, divisibility tests, the perpetual calendar which provides the day of the week of any date, round-robin tournaments,and computer hashing functions for data storage are discussed. Preface vtl Chapter 5 developsFermat's little theorem and Euler's theorem which give some important congruencesinvolving powers of integers. Also, Wilson's theorem which gives a congruencefor factorials is discussed.Primality and probabilistic primality tests based on these results are developed. Pseudoprimes, strong pseudoprimes, and Carmichael numbers which masquaradeas primesare introduced. Chapter 6 is concernedwith multiplicative functions and their properties. Special emphasisis devotedto the Euler phi-function,the sum of the divisors function, and the number of divisors function and explicit formulae are developed for these functions. Mersenne primes and perfect numbers are discussed. Chapter 7 gives a thorough discussionof applicationsof number theory to cryptology, starting with classical cryptology. Character ciphers based on modular arithmetic are described,as is cryptanalysisof theseciphers. Block ciphers based on modular arithmetic are also discussed. Exponentiation ciphers and their applications are described, including an application to electronic poker. The concept of a public-key cipher system is introduced and the RSA cipher is describedin detail. Knapsackciphers are discussed,as are applicationsof cryptographyto computerscience. Chapter 8 includesdiscussions of the order of an integer and of primitive roots. Indices, which are similar to logarithms, are introduced. Primality testing basedon primitive roots is described. The minimal universalexponent is studied. Pseudo-random numbers and means for generating them are discussed.An applicationto the splicingof telephonecables is also given. Chapter 9 covers quadratic residues and the famous law of quadratic reciprocity. The Legendreand Jacobi symbolsare introducedand algorithms for evaluating them are developed. Euler pseudoprimesand a probabilistic primality test are covered. An algorithm for electronically flipping coins is developed. Chapter l0 coversrational and irrational numbers,decimal representations of real numbers,and finite simplecontinued fractions of rational and irrational numbers. Special attention is paid to the continued fractions of the square roots of po"itive integers. Chapter 1l treats some nonlinear diophantine equations. Pythagorean triples are described. Fermat's last theorem is discussed. Finallv. Pell's equation is covered. vill Preface Problem Sets After each sectionof the text there is a problem set containingexercises of various levelsof difficulty. Each set containsproblems of a numerical nature; these should be done to developcomputational skills. The more theoretical and challenging problems should be done by studentsafter they have mastered the computationalskills. There are many more problemsin the text than can be realistically done in a course. Answers are provided at the end of the book for selectedexercises, mostly those having numerical answers. Computer Projects After each section of the text there is a selectionof computer projects that involve concepts or algorithms discussedin that section. Students can write their programs in any computer language they choose, using a home or personalcomputer, or a minicomputer or mainframe. I encouragestudents to use a structuredprogramming languagesuch as C, PASCAL, or PL/ 1, to do these projects. The projects can serve as good ways to motivate a student to learn a new computer language, and can give those students with strong computer science backgrounds interesting projects to tie together computer scienceand mathematics. Unsolved Problems In the text and in the problem setsunsolved questions in number theory are mentioned. Most of these problems have eluded solution for centuries. The reader is welcome to work on these questions,but should be forewarned