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  • 初等数论及其应用(英文版)(第6版)[平装]
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  • 作者:罗森(KennethH.Rosen)(作者)
  • 出版社:机械工业出版社;第1版(2010年9月1日)
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  • ISBN:9787111317982

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    《初等数论及其应用(英文版)(第6版)》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,《初等数论及其应用(英文版)(第6版)》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

    作者:(美国)罗森(Kenneth H.Rosen)

    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《离散数学及其应用》的中文版和影印版均已由机械工业出版社引进出版。

    目录

    list of symbols x
    what is number theory?
    1 the integers 5
    1.1 numbers and sequences 5
    1.2 sums and products 16
    1.3 mathematical induction 23
    1.4 the fibonacci numbers 30
    1.5 divisibility 36

    2 integer representations and operations 45
    2.1 representations of integers 45
    2.2 computer operations with integers 54
    2.3 complexity of integer operations 61
    3 primes and greatest common divisors 69
    3.1 prime numbers 70
    3.2 the distribution of primes 79
    3.3 greatest common divisors and their properties 93
    3.4 the euclidean algorithm 102
    3.5 the fundamental theorem of arithmetic 112
    3.6 factorization methods and the fermat numbers 127
    3.7 linear diophantine equations 137

    4 congruences 145
    4.1 introduction to congruences 145
    4.2 linear congruences 157
    4.3 the chinese remainder theorem 162
    4.4 solving polynomial congruences 171
    4.5 systems of linear congruences 178
    4.6 factoring using the pollard rho method 187

    5 applications of congruences 191
    5.1 divisibility tests 191
    5.2 the perpetual calendar 197
    5.3 round-robin tournaments 202
    5.4 hashing functions 204
    5.5 check digits 209

    6 some special congruences 217
    6.1 wilson's theorem and fermat's little theorem 217
    6.2 pseudoprimes 225
    6.3 euler's theorem 234

    7 multiplicative functions 239
    7.1 the euler phi-function 239
    7.2 the sum and number of divisors 249
    7.3 perfect numbers and mersenne primes 256
    7.4 misbius inversion 269
    7.5 partitions 277

    8 cryptology 291
    8.1 character ciphers 291
    8.2 block and stream ciphers 300
    8.3 exponentiation ciphers 318
    8.4 public key cryptography 321
    8.5 knapsack ciphers 331
    8.6 cryptographic protocols and applications 338

    9 primitive roots 347
    9.1 the order of an integer and primitive roots 347
    9.2 primitive roots for primes 354
    9.3 the existence of primitive roots 360
    9.4 discrete logarithms and index arithmetic 368
    9.5 primality tests using orders of integers and primitive roots 378
    9.6 universal exponents 385

    10 applications of primitive roots and the
    order of an integer 393
    10.1 pseudorandom numbers 393
    10.2 the eigamal cryptosystem 402
    10.3 an application to the splicing of telephone cables 408

    11 quadratic residues 415
    11.1 quadratic residues and nonresidues 416
    11.2 the law of quadratic reciprocity 430
    11.3 the jacobi symbol 443
    11.4 euler pseudoprimes 453
    11.5 zero-knowledge proofs 461

    12 decimal fractions and continued fractions 469
    12.1 decimal fractions 469
    12.2 finite continued fractions 481
    12.3 infinite continued fractions 491
    12.4 periodic continued fractions 503
    12.5 factoring using continued fractions 517

    13 some nonlinear diophantine equations 521
    13.1 pythagorean triples 522
    13.2 fermat's last theorem 530
    13.3 sums of squares 542
    13.4 pell's equation 553
    13.5 congruent numbers 560

    14 the gaussian integers 577
    14.1 gaussian integers and gaussian primes 577
    14.2 greatest common divisors and unique factorization 589
    14.3 gaussian integers and sums of squares 599
    appendix a axioms for the set of integers 605
    appendix b binomial coefficients 608
    appendix c using maple and mathematica for number theory 615
    c.1 using maple for number theory 615
    c.2 using mathematica for number theory 619
    appendix d number theory web links 624
    appendix e tables 626
    answers to odd-numbered exercises 641
    bibliography 721
    index of biographies 733
    index 735
    photo credits 752

    序言

    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.

    文摘

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    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!