初等數(shù)論及其應(yīng)用

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.

內(nèi)容概要

本書(shū)特色：    經(jīng)典理論與現(xiàn)代應(yīng)用相結(jié)合。通過(guò)豐富的實(shí)例和練習(xí)，將數(shù)論的應(yīng)用引入了更高的境界，同時(shí)更新并擴(kuò)充了對(duì)密碼學(xué)這一熱點(diǎn)論題的討論。    內(nèi)容與時(shí)俱進(jìn)。不僅融合了最新的研究成果和新的理論，而且還補(bǔ)充介紹了相關(guān)的人物傳記和歷史背景知識(shí)。    習(xí)題安排別出心裁。書(shū)中提供兩類(lèi)由易到難、富有挑戰(zhàn)的習(xí)題：一類(lèi)是計(jì)算題，另一類(lèi)是上機(jī)編程練習(xí)。這使得讀者能夠?qū)?shù)學(xué)理論與編程技巧實(shí)踐聯(lián)系起來(lái)。此外，本書(shū)在上一版的基礎(chǔ)上對(duì)習(xí)題進(jìn)行了大量更新和修訂。

作者簡(jiǎn)介

Kenneth H.Rosen，1972年獲密歇根大學(xué)數(shù)學(xué)學(xué)士學(xué)位，1976年獲麻省理工學(xué)院數(shù)學(xué)博士學(xué)位，1982年加入貝爾實(shí)驗(yàn)室，現(xiàn)為AT & T實(shí)驗(yàn)室特別成員，國(guó)際知名的計(jì)算機(jī)數(shù)學(xué)專(zhuān)家。Rosen博士對(duì)數(shù)論領(lǐng)域與數(shù)學(xué)建模領(lǐng)域頗有研究，并寫(xiě)過(guò)很多經(jīng)典論文及專(zhuān)著。他的經(jīng)典著作《離散數(shù)學(xué)及其應(yīng)

書(shū)籍目錄

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

章節(jié)摘錄

插圖：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！

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