本书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。

本书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。

本书特色：

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，本书在上一版的基础上对习题进行了大量更新和修订。

Preface

List of Symbols

What Is Number Theory?

1 The Integers

 1.1 Numbers and Sequences

 1.2 Sums and Products

 1.3 Mathematical Induction

 1.4 The Fibonacci Numbers

 1.5 Divisibility

2 Integer Representations and Operations

 2.1 Representations of Integers

 2.2 Computer Operations with Integers

 2.3 Complexity of Integer Operations

3 Primes and Greatest Common Divisors

 3.1 Prime Numbers

 3.2 The Distribution of Primes

 3.3 Greatest Common Divisors and their Properties

 3.4 The Euclidean Algorithm

 3.5 The Fundamental Theorem of Arithmetic

 3.6 Factorization Methods and the Fermat Numbers

 3.7 Linear Diophantine Equations

4 Congruences

 4.1 Introduction to Congruences

 4.2 Linear Congruences

 4.3 The Chinese Remainder Theorem

 4.4 Solving Polynomial Congruences

 4.5 Systems of Linear Congruences

 4.6 Factoring Using the Pollard Rho Method

5 Applications of Congruences

 5.1 Divisibility Tests

 5.2 The Perpetual Calendar

 5.3 Round-Robin Tournaments

 5.4 Hashing Functions

 5.5 Check Dieits

6 Some Special Congruences

 6.1 Wilson's Theorem and Fermat's Little Theorem

 6.2 Pseudoprimes

 6.3 Euler's Theorem

7 Multiplicative Functions

 7.1 The Euler Phi-Function

 7.2 The Sum and Number of Divisors

 7.3 Perfect Numbers and Mersenne Primes

 7.4 M6bius Inversion

 7.5 Partitions

8 Cryptology

 8.1 Character Ciphers

 8.2 Block and Stream Ciphers

 8.3 Exponentiation Ciphers

 8.4 Public Key Cryptography

 8.5 Knapsack Ciphers

 8.6 Cryptographic Protocols and Applications

9 Primitive Roots

 9.1 The Order of an Integer and Primitive Roots

 9.2 Primitive Roots for Primes

 9.3 The Existence of Primitive Roots

 9.4 Discrete Logarithms and Index Arithmetic

 9.5 Primality Tests Using Orders of Integers and Primitive Roots

 9.6 Universal Exponents

10 Applications of Primitive Roots and the

 Order of an Integer

 10.1 Pseudorandom Numbers

 10.2 The E1Gamal Cryptosystem

 10.3 An Application to the Splicing of Telephone Cables

11 Quadratic Residues

 11.1 Quadratic Residues and Nonresidues

 11.2 The Law of Quadratic Reciprocity

 11.3 The Jacobi Symbol

 11.4 Euler Pseudoprimes

 11.5 Zero-Knowledge Proofs

12 Decimal Fractions and Continued Fractions

 12.1 Decimal Fractions

 12.2 Finite Continued Fractions

 12.3 Infinite Continued Fractions

 12.4 Periodic Continued Fractions

 12.5 Factoring Using Continued Fractions

13 Some Nonlinear Diophantine Equations

 13.1 Pythagorean Triples

 13.2 Fermat's Last Theorem

 13.3 Sums of Squares

 13.4 Pell's Equation

 13.5 Congruent Numbers

14 The Gaussian Integers

 14.1 Gaussian Integers and Gaussian Primes

 14.2 Greatest Common Divisors and Unique Factorization

 14.3 Gaussian Integers and Sums of Squares

Appendix A Axioms for the Set of Integers

Appendix B Binomial Coefficients

Appendix C Using Maple and Mathematica for Number Theory

 C.1 Using Maple for Number Theory

 C.2 Using Mathematica for Number Theory

Appendix D Number Theory Web Links

Appendix E Tables

 Answers to Odd-Numbered Exercises

 Bibliography

 Index of Biographies

 Index

 Photo Credits