本书在阐述了数论丰富的历史的基础上，扩展了数论相关领域中的前沿知识与课题，既可以很好地与本科生的数论课程相契合，又适用于高中课程，是一本综合性、实用性较强的著作。本书共包含21章，其中既有关于费马大定理、椭圆曲线、斐波那契数、二项式定理、矩阵等问题的阐释，也有关于代数整数、高斯型整数、连分数、算术函数、数的几何、质数与因式分解、二次互反律等内容的具体例题与解答，还有RSA加密、Vigenere（维吉纳尔）密码、流密码、分块密码等密码学的前沿问题，为读者扩展数论与密码学方面的知识提供了丰富的资料。

Preface
1 Introduction
1.1 Diophantine Equations
1.2 Modular Arithmetic
1.3 Primes and the Distribution of Primes
1.4 Cryptography
2 Divisibility
2.1 Divisibility
2.2 Euclid's Theorem
2.3 Euclid's Original Proof
2.4 The Sieve of Eratosthenes
2.5 The Division Algorithm
2.5.1 A Cryptographic Application
2.6 The Greatest Common Divisor
2.7 The Euclidean Algorithm
2.7.1 The Extended Euclidean Algorithm
2.8 Other Bases
2.9 Fermat and Mersenne Numbers
2.10 Chapter Highlights
2.11 Problems
2.11.1 Exercises
2.11.2 Projects
2.11.3 Computer Explorations
2.11.4 Answers to "Check Your Understanding"
3 Linear Diophantine Equations
3.1 ax + by=c
3.2 The Postage Stamp Problem
3.3 Chapter Highlights
3.4 Problems
3.4.1 Exercises
3.4.2 Answers to "Check Your Understanding"
4 Unique Factorization
4.1 The Starting Point
4.2 The Fundamental Theorem of Arithmetic
4.3 Euclid and the Fundamental Theorem of Arithmetic
4.4 Chapter Highlights
4.5 Problems
4.5.1 Exercises
4.5.2 Projects
4.5.3 Answers to "Check Your Understanding"
5 Applications of Unique Factorization
5.1 A Puzzle
5.2 Irrationality Proofs
5.2.1 Four More Proofs That √2 Is Irrational
5.3 The Rational Root Theorem
5.4 Pythagorean Triples
5.5 Differences of Squares
5.6 Prime Factorization of Factorials
5.7 The Riemann Zeta Function
5.7.1 ∑1/p Diverges
5.8 Chapter Highlights
5.9 Problems
5.8.1 Exercises
5.9.2 Projects
5.9.3 Computer Explorations
……
6 Congruences
7 Classical Cryptosystems
8 Fermat, Euler, and Wilson
9 RSA
10 Polynomial Congruences
11 Order and Primitive Roots
12 More Cryptographic Applications
13 Quadratic Reciprocity
14 Primality and Factorization
15 Geometry of Numbers
16 Arithmetic Functions
17 Continued Fractions
18 Gaussian Integers
19 Algebraic Integers
20 The Distribution of Primes
21 Epilogue: Fermat's Last Theorem
A Supplementary Topics
B Answers and Hints for Odd-Numbered Exercises
Index
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