初等数论及其应用 第5版PDF电子书下载

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 CommonDivisors 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 M？bius 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 ElGamal 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-Knowledge Proofs 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 forNumberTheory 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