Elementary number theory and its applications = 初等数论及其应用 / 4th ed.
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ISBN：9787111138150
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简介
本书主要介绍数论的应用，包括整数、素数与最大公约数、同余变换、递增函数、密码学、本原根、二
次剩余与反比、小数与连分数、非线性丢番图方程等内容。
本书没有刻板的说教，而是以别致的方式，使学习数论变得轻松。此外，别出心裁的习题安排是本书的另一特色。每一节中都含有两类练习题，一类是笔答题，另一类是上机编程练习，这使得读者能够将书中的数学内容与实际的编程技巧联系起来。
本书自出版以来，深受读者好评，并已在数百所大学中被广泛采用。第4版除继承了前几版的优点之外，具备以下方面更加出色的特点：
通过大量实例将数论的应用引入了更高的境界。
更新并扩充了对密码学这一热点论题的介绍。
新增了关于默比乌斯反演与解多项式同余方程的内容。
证明更加严谨完善。
经过更加仔细的审核以确保内容的准确性。
给出了一些关于数论的网站地址。
目录
what is number theory? 1
1 the integers 5
1.1 numbers, sequences, and sums
1.2 mathematical induction 18
1.3 the fibonacci numbers 24
1.4 divisibility 31
2 integer representations and operations 39
2.1 representations of integers 39
2.2 computer operations with integers 49
2.3 complexity of integer operations 56
3 primes and greatest common divisors 65
3.1 prime numbers 66
3.2 greatest common divisors 80
3.3 the euclidean algorithm 86
3.4 the fundamental theorem of arithmetic 97
3.5 factorization methods and the fermat numbers 109
3.6 linear diophantine equations 119
4 congruences 127
4.1 introduction to congruences 127
4.2 linear congruences 139
.4.3 the chinese remainder theorem 143
4.4 solving polynomial congruences 153
4.5 systems of linear congruences 160
4.6 factoring using the pollard rho method 170
5 applications of congruences 173
5.1 divisibility tests 173
5.2 the perpetual calendar 179
5.3 round-robin tournaments 184
5.4 hashing functions 186
5.5 check digits 191
6 some special congruences 197
6.1 wilson's theorem and fermat's little theorem 197
6.2 pseudopfimes 205
6.3 euler's theorem 215
7 multiplicative functions 221
7.1 the euler phi-function 222
7.2 the sum and number of divisors 232
7.3 perfect numbers and mersenne primes 239
7.4 m6bius inversion 251
8 cryptology 259
8.1 character ciphers 260
8.2 block and stream ciphers 268
8.3 exponentiation ciphers 282
8.4 public-key cryptography 285
8.5 knapsack ciphers 292
8.6 cryptographic protocols and applications 299
9 primitive roots 307
9.1 the order of an integer and primitive roots 308
9.2 primitive roots for primes 315
9.3 the existence of primitive roots 321
9.4 index arithmetic 329
9.5 primality tests using orders of integers and primitive roots 339
9.6 universal exponents 346
10 applications of primitive roots and the
order of an integer 353
10.1 pseudorandom numbers 353
10.2 the elgamal cryptosystem 363
10.3 an application to the splicing of telephone cables 368
11 quadratic residues 375
11.1 quadratic residues and nonresidues 376
11.2 the law of quadratic reciprocity 392
11.3 the jacobi symbol 404
11.4 euler pseudoprimes 412
11.5 zero-knowledge proofs 421
12 decimal fractions and continued fractions 429
12.1 decimal fractions 429
12.2 finite continued fractions 442
12.3 infinite continued fractions 452
12.4 periodic continued fractions 463
12.5 factoring using continued fractions 477
13 some nonlinear diophantine equations 481
13.1 pythagorean triples 482
13.2 fermat's last theorem 487
13.3 sums of squares 495
13.4 pell's equation 505
appendix a axioms for the set of integers 515
appendix b binomial coefficients 519
appendix c using maple and mathematica for number
theory 527
c.1 using maple for number theory 527
c.2 using mathematica for number theory 530
appendix d number theory web links 535
appendix e tables 537
answers to odd-numbered exercises 553
bibliography 611
index of biographies 623
index 625
photo credits 638
1 the integers 5
1.1 numbers, sequences, and sums
1.2 mathematical induction 18
1.3 the fibonacci numbers 24
1.4 divisibility 31
2 integer representations and operations 39
2.1 representations of integers 39
2.2 computer operations with integers 49
2.3 complexity of integer operations 56
3 primes and greatest common divisors 65
3.1 prime numbers 66
3.2 greatest common divisors 80
3.3 the euclidean algorithm 86
3.4 the fundamental theorem of arithmetic 97
3.5 factorization methods and the fermat numbers 109
3.6 linear diophantine equations 119
4 congruences 127
4.1 introduction to congruences 127
4.2 linear congruences 139
.4.3 the chinese remainder theorem 143
4.4 solving polynomial congruences 153
4.5 systems of linear congruences 160
4.6 factoring using the pollard rho method 170
5 applications of congruences 173
5.1 divisibility tests 173
5.2 the perpetual calendar 179
5.3 round-robin tournaments 184
5.4 hashing functions 186
5.5 check digits 191
6 some special congruences 197
6.1 wilson's theorem and fermat's little theorem 197
6.2 pseudopfimes 205
6.3 euler's theorem 215
7 multiplicative functions 221
7.1 the euler phi-function 222
7.2 the sum and number of divisors 232
7.3 perfect numbers and mersenne primes 239
7.4 m6bius inversion 251
8 cryptology 259
8.1 character ciphers 260
8.2 block and stream ciphers 268
8.3 exponentiation ciphers 282
8.4 public-key cryptography 285
8.5 knapsack ciphers 292
8.6 cryptographic protocols and applications 299
9 primitive roots 307
9.1 the order of an integer and primitive roots 308
9.2 primitive roots for primes 315
9.3 the existence of primitive roots 321
9.4 index arithmetic 329
9.5 primality tests using orders of integers and primitive roots 339
9.6 universal exponents 346
10 applications of primitive roots and the
order of an integer 353
10.1 pseudorandom numbers 353
10.2 the elgamal cryptosystem 363
10.3 an application to the splicing of telephone cables 368
11 quadratic residues 375
11.1 quadratic residues and nonresidues 376
11.2 the law of quadratic reciprocity 392
11.3 the jacobi symbol 404
11.4 euler pseudoprimes 412
11.5 zero-knowledge proofs 421
12 decimal fractions and continued fractions 429
12.1 decimal fractions 429
12.2 finite continued fractions 442
12.3 infinite continued fractions 452
12.4 periodic continued fractions 463
12.5 factoring using continued fractions 477
13 some nonlinear diophantine equations 481
13.1 pythagorean triples 482
13.2 fermat's last theorem 487
13.3 sums of squares 495
13.4 pell's equation 505
appendix a axioms for the set of integers 515
appendix b binomial coefficients 519
appendix c using maple and mathematica for number
theory 527
c.1 using maple for number theory 527
c.2 using mathematica for number theory 530
appendix d number theory web links 535
appendix e tables 537
answers to odd-numbered exercises 553
bibliography 611
index of biographies 623
index 625
photo credits 638
Elementary number theory and its applications = 初等数论及其应用 / 4th ed.
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