埃弗里斯特编著的《数论导引（影印版）》是“国外数学名著系列”之一，从最初等的数论知识谈起，一直讲到解析数论、代数数论、椭圆曲线以及数论在密码理论中的应用等，涉及范围很广阔，而且内容并不肤浅。书中还有不少练习题，以及历史的评注等。可供数论及相关专业研究生、教师及科研人员等学习参考。

Introduction

1 A Brief History of Prime

 1.1 Euclid and Primes

 1.2 Summing Over the Primes

 1.3 Listing the Primes

 1.4 Fermat Numbers

 1.5 Primality Testing

 1.6 Proving the Fundamental Theorem of Arithmetic

 1.7 Euclid's Theorem Revisited

2 Diophantine Equations

 2.1 Pythagoras

 2.2 The Fundamental Theorem of Arithmetic in Other Contexts

 2.3 Sums of Squares

 2.4 Siegel's Theorem

 2.5 Fermat, Catalan, and Euler

3 Quadratic Diophantine Equations

 3.1 Quadratic Congruences

 3.2 Euler's Criterion

 3.3 The Quadratic Reciprocity Law

 3.4 Quadratic Rings

 3.5 Units in Z

 3.6 Quadratic Forms

4 Recovering the Fundamental Theorem of Arithmetic

 4.1 Crisis

 4.2 An Ideal Solution

 4.3 Fundamental Theorem of Arithmetic for Ideals

 4.4 The Ideal Class Group

5 Elliptic Curves

 5.1 Rational Points

 5.2 The Congruent Number Problem

 5.3 Explicit Formulas

 5.4 Points of Order Eleven

 5.5 Prime Values of Elliptic Divisibility Sequences

 5.6 Ramanujan Numbers and the Taxicab Problem

6 Elliptic Functions

 6.1 Elliptic Functions

 6.2 Parametrizing an Elliptic Curve

 6.3 Complex Torsion

 6.4 Partial Proof of Theorem 6.5

7 Heights

 7.1 Heights on Elliptic Curves

 7.2 Mordell's Theorem

 7.3 The Weak Mordell Theorem: Congruent Number Curve

 7.4 The Parallelogram Law and the Canonical Height

 7.5 Mahler Measure and the Naive Parallelogram Law

8 The Riemann Zeta Function

 8.1 Euler's Summation Formula

 8.2 Multiplicative Arithmetic Functions

 8.3 Dirichlet Convolution

 8.4 Euler Products

 8.5 Uniform Convergence

 8.6 The Zeta Function Is Analytic

 8.7 Analytic Continuation of the Zeta Function

9 The Functional Equation of the Riemann Zeta Function

 9.1 The Gamma Function

 9.2 The Functional Equation

 9.3 Fourier Analysis on Schwartz Spaces

 9.4 Fourier Analysis of Periodic Functions

 9.5 The Theta Function

 9.6 The Gamma Function Revisited

10 Primes in an Arithmetic Progression

 10.1 A New Method of Proof

 10.2 Congruences Modulo 3

 10.3 Characters of Finite Abelian Groups

 10.4 Dirichlet Characters and L-Functions

 10.5 Analytic Continuation and Abel's Summation Formula

 10.6 Abel's Limit Theorem

11 Converging Streams

 11.1 The Class Number Formula

 11.2 The Dedekind Zeta Function

 11.3 Proof of the Class Number Formula

 11.4 The Sign of the Gauss Sum

 11.5 The Conjectures of Birch and Swinnerton-Dyer

12 Computational Number Theory

 12.1 Complexity of Arithmetic Computations

 12.2 Public-key Cryptography

 12.3 Primality Testing: Euclidean Algorithm

 12.4 Primality Testing: Pseudoprimes

 12.5 Carmichael Numbers

 12.6 Probabilistic Primality Testing

 12.7 The Agrawal-Kayal-Saxena Algorithm

 12.8 Factorizing

 12.9 Complexity of Arithmetic in Finite Fields

References

Index