初等数论及其应用

出版时间：2010-9  出版社：机械工业出版社  作者：罗森  页数：752  
Tag标签：无  

前言

My goal in writing this text has been to write an accessible and inviting introduction to number theory. Foremost, I wanted to create an effective tool for teaching and learning.I hoped to capture the richness and beauty of the subject and its unexpected usefulness.Number theory is both classical and modem, and, at the same time, both pure and applied. In this text, I have strived to capture these contrasting aspects of number theory. I have worked hard to integrate these aspects into one cohesive text. This book is ideal for an undergraduate number theory course at any level. No formal prerequisites beyond college algebra are needed for most of the material, other than some level of mathematical maturity. This book is also designed to be a source book for elementary number theory; it can serve as a useful supplement for computer science courses and as a primer for those interested in new developments in number theory and cryptography. Because it is comprehensive, it is designed to serve both as a textbook and as a lifetime reference for elementary number theory and its wide-ranging applications. This edition celebrates the silver anniversary of this book. Over the past 25 years,close to 100,000 students worldwide have studied number theory from previous editions.Each successive edition of this book has benefited from feedback and suggestions from many instructors, students, and reviewers. This new edition follows the same basic approach as all previous editions, but with many improvements and enhancements. I invite instructors unfamiliar with this book, or who have not looked at a recent edition, to carefully examine the sixth edition. I have confidence that you will appreciate the rich exercise sets, the fascinating biographical and historical notes, the up-to-date coverage, careful and rigorous proofs, the many helpful examples, the rich applications, the support for computational engines such as Maple and Mathematica, and the many resources available on the Web.

内容概要

本书特色：    经典理论与现代应用相结合。通过丰富的实例和练习，将数论的应用引入了更高的境界，同时更新并扩充了对密码学这一热点论题的讨论。    内容与时俱进。不仅融合了最新的研究成果和新的理论，而且还补充介绍了相关的人物传记和历史背景知识。    习题安排别出心裁。书中提供两类由易到难、富有挑战的习题：一类是计算题，另一类是上机编程练习。这使得读者能够将数学理论与编程技巧实践联系起来。此外，本书在上一版的基础上对习题进行了大量更新和修订。

作者简介

Kenneth H.Rosen，1972年获密歇根大学数学学士学位，1976年获麻省理工学院数学博士学位，1982年加入贝尔实验室，现为AT & T实验室特别成员，国际知名的计算机数学专家。Rosen博士对数论领域与数学建模领域颇有研究，并写过很多经典论文及专著。他的经典著作《离散数学及其应

书籍目录

PrefaceList of SymbolsWhat Is Number Theory?1  The Integers  1.1 Numbers and Sequences  1.2 Sums and Products  1.3 Mathematical Induction  1.4 The Fibonacci Numbers  1.5 Divisibility2  Integer Representations and Operations  2.1 Representations of Integers  2.2 Computer Operations with Integers  2.3 Complexity of Integer Operations3  Primes and Greatest Common Divisors  3.1 Prime Numbers  3.2 The Distribution of Primes  3.3 Greatest Common Divisors and their Properties  3.4 The Euclidean Algorithm  3.5 The Fundamental Theorem of Arithmetic  3.6 Factorization Methods and the Fermat Numbers  3.7 Linear Diophantine Equations4  Congruences  4.1 Introduction to Congruences  4.2 Linear Congruences  4.3 The Chinese Remainder Theorem  4.4 Solving Polynomial Congruences  4.5 Systems of Linear Congruences  4.6 Factoring Using the Pollard Rho Method5  Applications of Congruences  5.1 Divisibility Tests  5.2 The Perpetual Calendar  5.3 Round-Robin Tournaments  5.4 Hashing Functions  5.5 Check Dieits6  Some Special Congruences  6.1 Wilson's Theorem and Fermat's Little Theorem  6.2 Pseudoprimes  6.3 Euler's Theorem7  Multiplicative Functions  7.1 The Euler Phi-Function  7.2 The Sum and Number of Divisors  7.3 Perfect Numbers and Mersenne Primes  7.4 M6bius Inversion  7.5 Partitions8  Cryptology  8.1 Character Ciphers  8.2 Block and Stream Ciphers  8.3 Exponentiation Ciphers  8.4 Public Key Cryptography  8.5 Knapsack Ciphers  8.6 Cryptographic Protocols and Applications9  Primitive Roots  9.1 The Order of an Integer and Primitive Roots  9.2 Primitive Roots for Primes  9.3 The Existence of Primitive Roots  9.4 Discrete Logarithms and Index Arithmetic  9.5 Primality Tests Using Orders of Integers and Primitive Roots  9.6 Universal Exponents10  Applications of Primitive Roots and the  Order of an Integer  10.1 Pseudorandom Numbers  10.2 The E1Gamal Cryptosystem  10.3 An Application to the Splicing of Telephone Cables11  Quadratic Residues  11.1 Quadratic Residues and Nonresidues  11.2 The Law of Quadratic Reciprocity  11.3 The Jacobi Symbol  11.4 Euler Pseudoprimes  11.5 Zero-Knowledge Proofs12  Decimal Fractions and Continued Fractions  12.1 Decimal Fractions  12.2 Finite Continued Fractions  12.3 Infinite Continued Fractions  12.4 Periodic Continued Fractions  12.5 Factoring Using Continued Fractions13  Some Nonlinear Diophantine Equations  13.1 Pythagorean Triples  13.2 Fermat's Last Theorem  13.3 Sums of Squares  13.4 Pell's Equation  13.5 Congruent Numbers14  The Gaussian Integers  14.1 Gaussian Integers and Gaussian Primes  14.2 Greatest Common Divisors and Unique Factorization  14.3 Gaussian Integers and Sums of SquaresAppendix A  Axioms for the Set of IntegersAppendix B  Binomial CoefficientsAppendix C  Using Maple and Mathematica for Number Theory  C.1 Using Maple for Number Theory  C.2 Using Mathematica for Number TheoryAppendix D  Number Theory Web LinksAppendix E  Tables  Answers to Odd-Numbered Exercises  Bibliography  Index of Biographies  Index  Photo Credits

章节摘录

插图：Experimentation and exploration play a key role in the study of number theory. Theresults in this book were found by mathematicians who often examined large amounts ofnumerical evidence, looking for patterns and making conjectures. They worked diligentlyto prove their conjectures; some of these were proved and became theorems, others wererejected when counterexamples were found, and still others remain unresolved. As youstudy number theory, I recommend that you examine many examples, look for patterns,and formulate your own conjectures. You can examine small examples by hand, much asthe founders of number theory did, but unlike these pioneers, you can also take advantageof today's vast computing power and computational engines. Working through examples,either by hand or with the aid of computers, will help you to learn the subject——and youmay even find some new results of your own！

用户评论 (总计9条)

•   数论其实在密码学或者另外一些工程上都能看到一些身影（比如众所周知的中国余数定理在好多地方都能见到）。书的内容也很全面。是数学爱好者必收的书，然后喜欢研究程序算法的大概也能从中吸取很多营养~
•   Rosen的书，木话说，好书！我从第4版买到第6版
•   内容详实，逻辑清晰，比国内的教材更充实易懂，适合数学爱好者
•   今天刚收到这本书，无论是从纸张还是内容都让人感觉很舒服
•   很多著作的水平因译者水平而异，还是看原版来得明了，收藏了。
•   我的最爱 收藏了
•   收到书和以前一样快，但是还没读
•   感觉这本书比北大潘老师写的要温暖些，推理上更亲切些。当然潘老师的书早已作为初等数论的精典教材为大家所熟知。两本书结合来看，效果更好。
•   书的内容不错，就是书的纸张不好。