初等数论及其应用

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 wilsons theorem and fermats little theorem 217

6.2 pseudoprimes 225

6.3 eulers 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 fermats last theorem 530

13.3 sums of squares 542

13.4 pells 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

《初等数论及其应用（英文版）（第6版）》是数论课程的经典教材，自出版以来，深受读者好评，被美国加州大学伯克利分校、伊利诺伊大学、得克萨斯大学等数百所名校采用。
《初等数论及其应用（英文版）（第6版）》以经典理论与现代应用相结合的方式介绍了初等数论的基本概念和方法，内容包括整除、同余、二次剩余、原根以及整数的阶的讨论和计算。

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。