(特价书)初等数论及其应用(英文影印版)(第6版)

.　　Greatest common divisors are now defined in the first chapter, as is what it means for two integers to be relatively prime. The term Bezout coefficients is now introduced and used in the book.
　　~ Jacobi symbols
　　More motivation is provided for the usefulness of Jacobi symbols. In particular, an expanded discussion on the usefulness of the Jacobi symbol in evaluating Legendre symbols is now provided.
　　~ Enhanced exercise sets
　　Extensive work has been done to improve exercise sets even farther. Several hundred new exercises, ranging from routine to challenging, have been added. Moreover, new computational and exploratory exercises can be found in this new edition.
　　~ Accurancy
　　More attention than ever before has been paid to ensuring the accuracy of this edition. Two independent accuracy checkers have examined the entire text and the answers to exercises.
　　~ Web Site, www. pearsonhighered, com/rosen
　　The Web site for this edition has been considerably expanded. Students and instructors will find many new resources they can use in conjunction with the book. Among the new features are an expanded collection of applets, a manual for using comptutional engines to explore number theory, and a Web page devoted to number theory news.
　　Exercise Sets
　　Because exercises are so important, a large percentage of my writing and revision work has been devoted to the exercise sets. Students should keep in mind that the best way to learn mathematics is to work as many exercises as possible. I will briefly describe the types of exercises found in this book and where to find answers and solutions.
　　~ Standard Exercises
　　Many routine exercises are included to develop basic skills, with care taken so that both odd-numbered and even-numbered exercises of this type are included. A large number of intermediate-level exercises help students put several concepts together to form new results. Many other exercises and blocks of exercises are designed to develop new concepts.
　　~ Exercise Legend
　　Challenging exercises are in ample supply and are marked with one star (*) indicating a difficult exercise and two stars (**) indicating an extremely difficult exercise. There are some exercises that contain results used later in the text; these are marked with a arrow symbol (>). These exercises should be assigned by instructors whenever possible.
　　~ Exercise Answers
　　The answers to all odd-numbered exercises are provided at the end of the text. More complete solutions to these exercises can be found in the Student's Solutions Manual that can be found on the Web site for this book. All solutions have been carefully checked and rechecked to ensure accuracy.
　　~ Computational Exercises
　　Each section includes computations and explorations designed to be done with a computational program, such as Maple, Mathematica, PARI/GP, or Sage, or using programs written by instructors and/or students. There are routine computational exercises students can do to learn how to apply basic commands (as described in Appendix D for Maple and Mathematica and on the Web site for PARI/GP and Sage), as well as more open-ended questions designed for experimentation and creativity. Each section also includes a set of programming projects designed to be done by students using a programming language or the computational program of their choice. The Student's Manual to Computations and Explorations on the Web site provides answers, hints, and guidance that will help students use computational tools to attack these exercises.
　　Web Site
　　Students and instructors will find a comprehensive collection of resources on this book's Web site. Students (as well as instructors) can find a wide range of resources at www.pearsonhighered.com/rosen. Resources intended for only instructor use can be accessed at www.pearsonhighered.com/irc; instructors can obtain their password for these resources from Pearson.
　　~ External Links
　　The Web site for this book contains a guide providing annotated links to many Web sites relevant to number theory. These sites are keyed to the page in the book where relevant material is discussed. These locations are marked in the book with the icon . For convenience, a list of the most important Web sites related to number theory is provided in Appendix D.
　　~ Number Theory News
　　The Web site also contains a section highlighting the latest discoveries in number theory.
　　~ Student's Solutions Manual
　　Worked-out solutions to all the odd-numbered exercises in the text and sample exams can be found in the online Student's Solution Manual.
　　~ Student's Manual for Computations and Explorations
　　A manual providing resources supporting the computations and explorations can be found on the Web site for this book. This manual provides worked-out solutions or partial solutions to many of these computational and exploratory exercises, as well as hints and guidance for attacking others. This manual will support, to varying degrees, different comptutional environments, including Maple, Mathematica, and PARI/GP.
　　~ Applets
　　An extensive collection of applets are provided on the Web site. These applets can be used by students for some common computations in number theory and to help understand concepts and explore conjectures. Besides algorithms for comptutions in number theory, a collection of cryptographic applets is also provided. These include applets for encyrption, decryption, cryptanalysis, and cryptographic protocols, adderssing both classical ciphers and the RSA cryptosystem. These cryptographic applets can be used for individual, group, and classroom activities.
　　~ SuggestedProjects
　　A useful collection of suggested projects can also be found on the Web site for this book. These projects can serve as final projects for students and for groups of students.
　　~ Instructor's Manual
　　Worked solutions to all exercises in the text, including the even-numbered execises, and a variety of other resources can be found on the Web site for instructors (which is not available to students). Among these other resources are sample syllabi, advice on planning which sections to cover, and a test bank.
　　How to Design a Course Using this Book
　　This book can serve as the text for elementary number theory courses with many different slants and at many different levels. Consequently, instructors will have a great deal of flexibility designing their syllabi with this text. Most instructors will want to cover the core material in Chapter 1 (as needed), Section 2.1 (as needed), Chapter 3, Sections 4.1--4.3, Chapter 6, Sections 7.1-7.3, and Sections 9.1-9.2.
　　To fill out their syllabi, instructors can add material on topics of interest. Generally, topics can be broadly classified as pure versus applied. Pure topics include M6bius inversion (Section 7.4), integer partitions (Section 7.5), primitive roots (Chapter 9), continued fractions (Chapter 12), diophantine equations (Chapter 13), and Guassian integers (Chapter 14).
　　Some instructors will want to cover accessible applications such as divisibility tests, the perpetual calendar, and check digits (Chapter 5). Those instructors who want to stress computer applications and cryptography should cover Chapter 2and Chapter 8. They may also want to include Sections 9.3 and 9.4, Chapter 10, and Section 11.5.
　　 After deciding which topics to cover, instructors may wish to consult the following figure displaying the dependency of chapters:
　　 Although Chapter 2 may be omitted if desired, it does explain the big-O notation used throughout the text to describe the complexity of algorithms. Chapter 12 depends only on Chapter 1, as shown, except for Theorem 12.4, which depends on material from Chapter 9. Section 13.4 is the only part of Chapter 13 that depends on Chapter 12. Chapter 11 can be studied without covering Chapter 9 if the optional comments involving primitive roots in Section 9.1 are omitted. Section 14.3 should also be covered in conjunction with Section 13.3.
　　 For further assistance, instructors can consult the suggested syllabi for courses with different emphases provided in the Instructor's Resource Guide on the Web site.
　　Acknowledgments
　　I appreciate the continued strong support and enthusiam of Bill Hoffman, my editor at Pearson and Addison-Wesley far longer than any of the many editors who have preceded him, and Greg Tobin, president of the mathematics division of Pearson. My special gratitude goes to Caroline Celano, associate editor, for all her assistance with development and production of the new edition. My appreciation also goes to the production, marketing, and media team behind this book, including Beth Houston (Production Project Manager), Maureen Raymond (Photo Researcher), Carl Cottrell (Media Producer), Jeff Weidenaar (Executive Marketing Manager), Kendra Bassi (Marketing Assistant), and Beth Paquin (Designer) at Pearson, and Paul Anagnostopoulos (project manager), Jacqui Scarlott (composition), Rick Camp (copyeditor and proofreader), and Laurel Muller (artist) at Windfall Software. I also want to reiterate my thanks to all those who supported my work on the first five editions of this book, including the multitude of my previous'editors at Addison Wesley and my management at AT&T Bell Laboratories (and its various incarnations).
　　 Special thanks go to Bart Goddard who has prepared the solutions of all exercises in this book, including those found at the end of the book and on the Web site, and who has reviewed the entire book. I am also grateful to Jean-Claude Evard and Roger Lipsett for their help checking and rechecking the entire manuscript, including the answers to exercises. I would also like to thank David Wright for his many contributions to the Web site for this book, including material on PARI/GP, number theory and cryptography applets, the computation and exploration manual, and the suggested projects. Thanks also goes to Larry Washington and Keith Conrad for their helpful suggestions concerning congruent numbers and elliptic curves.
　　Reviewers
　　I have benefited from the thoughtful reviews and suggestions from users of previous editions, to all of whom I offer heartfelt thanks. Many of their ideas have been incorporated in this edition. My profound thanks go to the reviewers who helped me prepare the sixth edition:
　　 Jennifer Beineke, Western New England College
　　 David Bradley, University of Maine-Orono
　　 Flavia Colonna, George Mason University
　　 Keith Conrad, University of Connecticut
　　 Pavel Guerzhoy, University of Hawaii
　　 Paul E. Gunnells, University of Massachusetts-Amherst
　　 Charles Parry, Virginia Polytechnic Institute and State University
　　 Holly Swisher, Oregon State University
　　 Lawrence Sze, California State Polytechnic University, Pomona
　　 I also wish to thank again the approximately 50 reviewers of previous editions of this book. They have helped improve this book throughout its life. Finally, I thank in advance all those who send me suggestions and corrections in the future. You may send such material to me in care of Pearson at math@pearson.com.
　　Kenneth H. Rosen
　　Middletown, New Jersey