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The iteration of the transformation T n: The Collatz conjecture assertsthat with any Magic Pro MP-A2N-8M point, Magic Pro MP-A2N-8M iteration of? Elementary number theory and its applications. R67 rsBN I. 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.

Elementary number theory and its applications

Published simultaneously in Canada. It is a classical subject and has a reputation for being the Magic Pro MP-A2N-8M 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 course at 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 designed to 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 Magic Pro MP-A2N-8M their computationai complexity, computer arithmetic with large integers, binary and hexadecimal representations of integers, primality testing, pseudoprimality,pseudo-randomnumbers, 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 Sections Magic Pro MP-A2N-8M.

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Mersenne primes, and perfect numbers; some of this material is used in Chapter 8. Chapter 7 covers the applications of number theory to cryptology.

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Chapter 8 deals Magic Pro MP-A2N-8M primitive roots; Sections 8. Most instructors will want to include Section 8. The Contents The reader can determine which chapters to study based on the following descriptionof their contents. Chapter I introduces two importants tools in establishing results about the integers, the well-ordering property and the principle of mathematical induction.

Elementary number theory and its applications

Recursive definitions 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, Magic Pro MP-A2N-8M 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 Magic Pro MP-A2N-8M is described. 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 primes are 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 Magic Pro MP-A2N-8M are discussed.


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