简介
《初等数论及其应用（英文版·第5版）》的核心是以一种有助于理解和引人入胜的方式阐述经典初等论，关键结果的史料和重要性得到记述，在精心开展每个论题的基本材料之后，接着论述同一论更复杂的结果，本书的主要长处是包括了数论的种种应用，一旦需要的理论得以建立，应用就以灵活的方式编入教材，应用设计成有助于促进理论的扩展和阐明初等数论在不同方面的用处，数论广泛应用于密码学，经典密码、分组密码及序列密、公钥密码系统和密码协议都被包括在内，对计算机科学的其他应用包括整数的快速乘法、伪随机数及校验数字，对于许多其他领域的应用，例如调度、电话、昆虫学和动物学，也可在教材中找到。
本教材包括极为广泛和多种多样的习题，收入许多常规习题是为了训练基本技能，已注意将带有奇数编号和偶数编号的两种习题包含在这一类题中，大量中等难度的题有助于学生把若干概念结合起来形成新的结果，许多其他习题或习题组则是为发展新概念而设计的，具有挑战性的习题也是充足的，用单星号表示难题，双星号表示很难的题，有的题包含以后正文中要用到的结果，这些题用手指符号表示，这样的习题教师在适当的时候应尽可能布置。
本书中包括数论的最新发现，描述了许多未解决问题的现状，例如新的理论成果，2004年9月关于素数和因数分解的新发现已列入这一版的第一次印刷之中，这些发现将有助于读者理解数论产一个极为活跃的研究领域，他们可以看到甚至他们自己有可能参与发现新的素数。
目录
what is number theory? 1
1 the integers 5
1.1 numbers and sequences 6
1.2 sums and products 16
1.3 mathematical induction 23
1.4 the fibonacci numbers 30
1.5 divisibility 37
2 integer representations and operations 43
2.1 representations of integers 43
2.2 computer operations with integers 53
2.3 complexity of integer operations 60
3 primes and greatest common divisors 67
3.1 prime numbers 68
3.2 the distribution of primes 77
3.3 greatest common divisors 90
3.4 the euclidean algorithm 97
3.5 the fundamental theorem of arithmetic 108
3.6 factorization methods and the fermat numbers 123
3.7 linear diophantine equations 133
4 congruences 141
.4.1 introduction to congruences 141
4.2 linear congruences 153
4.3 the chinese remainder theorem 158
4.4 solving polynomial congruences 168
4.5 systems of linear congruences 174
4.6 factoring using the pollard rho method 184
5 applications of congruences 189
5.1 divisibility tests 189
5.2 the perpetual calendar 195
5.3 round-robin tournaments 200
5.4 hashing functions 202
5.5 check digits 207
6 some special congruences 215
6.1 wilson's theorem and fermat's little theorem 215
6.2 pseudoprimes 223
6.3 euler's theorem 233
7 multiplicative functions 239
7.1 the euler phi-function 239
7.2 the sum and number of divisors 250
7.3 perfect numbers and mersenne primes 257
7.4 mtbius inversion 269
8 cryptology 277
8.1 character ciphers 278
8.2 block and stream ciphers 286
8.3 exponentiation ciphers 305
8.4 public key cryptography 308
8.5 knapsack ciphers 316
8.6 cryptographic protocols and applications 323
9 primitive roots 333
9.1 the order of an integer and primitive roots 334
9.2 primitive roots for primes 341
9.3 the existence of primitive roots 347
9.4 index arithmetic 355
9.5 primality tests using orders of integers and primitive roots 365
9.6 universal exponents 372
10 applications of primitive roots and the order of an integer 379
10.1 pseudorandom numbers 379
10.2 the e1gamal cryptosystem 389
10.3 an application to the splicing of telephone cables 394
11 quadratic residues 401
11.1 quadratic residues and nonresidues 402
11.2 the law of quadratic reciprocity 417
11.3 the jacobi symbol 430
11.4 euler pseudoprimes 439
11.5 zero-knowledgeproofs 448
12 decimal fractions and continued fractions 455
12.1 decimal fractions 455
12.2 finite continued fractions 468
12.3 infinite continued fractions 478
12.4 periodic continued fractions 490
12.5 factoring using continued fractions 504
13 some nonlinear diophantine equations 509
13.1 pythagorean triples 510
13.2 fermat's last theorem 516
13.3 sums of squares 528
13.4 pell's equation 539
14 the gaussian integers 547
14.1 gaussian integers and gaussian primes 547
14.2 greatest common divisors and unique factorization 559
14.3 gaussian integers and sums of squares 570
a axioms for the set of integers 577
b binomial coefficients 581
c using maple and mathematica for number theory 589
c.1 using maple for number theory 589
c.2 using mathematica for number theory 593
d number theory web links 599
e tables 601
answers to odd-numbered exercises 617
bibliography 689
index of biographies 703
index 705
photo credits 721
1 the integers 5
1.1 numbers and sequences 6
1.2 sums and products 16
1.3 mathematical induction 23
1.4 the fibonacci numbers 30
1.5 divisibility 37
2 integer representations and operations 43
2.1 representations of integers 43
2.2 computer operations with integers 53
2.3 complexity of integer operations 60
3 primes and greatest common divisors 67
3.1 prime numbers 68
3.2 the distribution of primes 77
3.3 greatest common divisors 90
3.4 the euclidean algorithm 97
3.5 the fundamental theorem of arithmetic 108
3.6 factorization methods and the fermat numbers 123
3.7 linear diophantine equations 133
4 congruences 141
.4.1 introduction to congruences 141
4.2 linear congruences 153
4.3 the chinese remainder theorem 158
4.4 solving polynomial congruences 168
4.5 systems of linear congruences 174
4.6 factoring using the pollard rho method 184
5 applications of congruences 189
5.1 divisibility tests 189
5.2 the perpetual calendar 195
5.3 round-robin tournaments 200
5.4 hashing functions 202
5.5 check digits 207
6 some special congruences 215
6.1 wilson's theorem and fermat's little theorem 215
6.2 pseudoprimes 223
6.3 euler's theorem 233
7 multiplicative functions 239
7.1 the euler phi-function 239
7.2 the sum and number of divisors 250
7.3 perfect numbers and mersenne primes 257
7.4 mtbius inversion 269
8 cryptology 277
8.1 character ciphers 278
8.2 block and stream ciphers 286
8.3 exponentiation ciphers 305
8.4 public key cryptography 308
8.5 knapsack ciphers 316
8.6 cryptographic protocols and applications 323
9 primitive roots 333
9.1 the order of an integer and primitive roots 334
9.2 primitive roots for primes 341
9.3 the existence of primitive roots 347
9.4 index arithmetic 355
9.5 primality tests using orders of integers and primitive roots 365
9.6 universal exponents 372
10 applications of primitive roots and the order of an integer 379
10.1 pseudorandom numbers 379
10.2 the e1gamal cryptosystem 389
10.3 an application to the splicing of telephone cables 394
11 quadratic residues 401
11.1 quadratic residues and nonresidues 402
11.2 the law of quadratic reciprocity 417
11.3 the jacobi symbol 430
11.4 euler pseudoprimes 439
11.5 zero-knowledgeproofs 448
12 decimal fractions and continued fractions 455
12.1 decimal fractions 455
12.2 finite continued fractions 468
12.3 infinite continued fractions 478
12.4 periodic continued fractions 490
12.5 factoring using continued fractions 504
13 some nonlinear diophantine equations 509
13.1 pythagorean triples 510
13.2 fermat's last theorem 516
13.3 sums of squares 528
13.4 pell's equation 539
14 the gaussian integers 547
14.1 gaussian integers and gaussian primes 547
14.2 greatest common divisors and unique factorization 559
14.3 gaussian integers and sums of squares 570
a axioms for the set of integers 577
b binomial coefficients 581
c using maple and mathematica for number theory 589
c.1 using maple for number theory 589
c.2 using mathematica for number theory 593
d number theory web links 599
e tables 601
answers to odd-numbered exercises 617
bibliography 689
index of biographies 703
index 705
photo credits 721
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