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这本书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。

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• 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[*],d>0
  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