Notation

1 Divisibility

1.1 Introduction

1.2 Divisibility

1.3 Primes

1.4 The Binomial Theorem

Notes on Chapter 1

2 Congruences

2.1 Congruences

2.2 Solutions of Congruences

2.3 The Chinese Remainder Theorem

2.4 Techniques of Numerical Calculation

2.5 Public-Key Cryptography

2.6 Prime Power Moduli

2.7 Prime Modulus

2.8 Primitive Roots and Power Residues

2.9 Congruences of Degree Two, Prime Modulus

2.10 Number Theory from an Algebraic Viewpoint

2.11 Groups, Rings, and Fields

Notes on Chapter 2

3 Quadratic Reciprocity and Quadratic Forms

3.1 Quadratic Residues

3.2 Quadratic Reciprocity

3.3 The Jacobi Symbol

3.4 Binary Quadratic Forms

3.5 Equivalence an Reduction of Binary Quadratic Forms

3.6 Sums of Two Squares

3.7 Positive Definite Binary Quadratic Forms

Notes on Chapter 3

4 Some Functions of Number Theory

4.1 Greatest Integer Function

4.2 Arithmetic Functions

4.3 The Möbius Inversion Formula

4.4 Recurrence Functions

4.5 Combinatorial Number Theory

Notes on Chapter 4

5 Some Diophantine Equations

5.1 The equation ax + by = c

5.2 Simultaneous Linear Equations

5.3 Pythagorean Triangles

5.4 Assorted Examples

5.5 Ternary Quadratic Forms

5.6 Rational Points on Curves

5.7 Elliptic Curves

5.8 Factorization Using Elliptic Curves

5.9 Curves of Genus Greater Than 1

Notes on Chapter 5

6 Farey Fractions and Irrational Numbers

6.1 Farey Sequences

6.2 Rational Approximations

6.3 Irrational Numbers

6.4 The Geometry of Numbers

Notes on Chapter 6

7 Simple Continued Fractions

7.1 The Euclidean Algorithm

7.2 Uniqueness

7.3 Infinite Continued Fractions

7.4 Irrational Numbers

7.5 Approximation to Irrational Numbers

7.6 Best Possible Approximations

7.7 Periodic Continued Fractions

7.8 Pell's Equation

7.9 Numerical Computation

Notes on Chapter 7

8 Primes and Multiplicative Number Theory

8.1 Elementary Prime Number Estimates

8.2 Dirichlet Series

8.3 Estimates of Arithmetic Functions

8.4 Primes in Arithmetic Progressions

Notes on Chapter 8

9 Algebraic Numbers

9.1 Polynomials

9.2 Algebraic Numbers

9.3 Algebraic Number Fields

9.4 Algebraic Integers

9.5 Quadratic Fields

9.6 Units in Quadratic Fields

9.7 Primes in Quadratic Fields

9.8 Unique Factorization

9.9 Primes in Quadratic Fields Having the Unique Factorization Property

9.10 The Equation x^{3} + y^{3} = z^{3}

Notes on Chapter 9

10 The Partition Function

10.1 Partitions

10.2 Ferrers Graphs

10.3 Formal Power Series, Generating Functions, and Euler's Identity

10.4 Euler's Formula; Bounds on p(n)

10.5 Jacobi's Formula

10.6 A Divisibility Property

Notes on Chapter 10

11 The Density of Sequences of Integers

11.1 Asymptotic Density

11.2 Schnirelmann Density and the αβ Theorem

Notes on Chapter 11

Appendices

A.1 The Fundamental Theorem of Algebra

A.2 Symmetric Functions

A.3 A Special Value of the Riemann Zeta Function

A.4 Linear Recurrences

General References

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Index