Preface to the second edition

Introduction

1. The fundamental theorem of arithmetic

Division algorithm

Greatest common divisor and Euclidean algorithm

Unique factorisation into primes

Infinity of primes

Mersenne primes

Summary

Historical note

Notes and answers

2. Modular addition and Euler’s φ function

Congruence classes and the Chinese remainder theorem

The groups (**Z**_{n}, +) and their generators

Euler’s φ function

Summing Euler’s function over divisors

Summary

Historical note

Notes and answers

3. Modular multiplication

Fermat’s theorem

Wilson’s theorem

Linear congruences

Fermat-Euler theorem

Simultaneous linear congruences

Lagrange’s theorem for polynomials

Primitive roots

Chevalley’s theorem

RSA codes

Summary

Historical note

Notes and answers

4. Quadratic residues

Quadratic residues and the Legendre symbol

Gauss’ lemma

Law of quadratic reciprocity

Summary

Historical note

Notes and answers

5. The equation x^{n} + y^{n} = z^{n}, for n = 2, 3, 4

The equation x^{2} + y^{2} = z^{2}

The equation x^{4} + y^{4} = z^{4}

The equation x^{2} + y^{2} + z^{2} = t^{2}

The equation x^{3} + y^{3} = z^{3}

Historical note

Notes and answers

6. Sums of squares

Sums of two squares

Sums of four squares

Sums of three squares

Triangular numbers

Historical note

Notes and answers

7. Partitions

Ferrers’ graphs

Generating functions

Euler’s theorem

Summary

Historical note

Notes and answers

8. Quadratic forms

Unimodular transformations

Equivalent quadratic forms

Discriminant

Proper representation

Reduced forms

Automorphs of definite quadratic forms

Summary

Historical note

Notes and answers

9. Geometry of numbers

Subgroups of a square lattice

Minkowski’s theorem in two dimensions

Subgroups of a cubic lattice

Minkowski’s theorem in three dimensions

Legendre’s theorem on ax^{2} + by^{2} + cz^{2} = 0

Summary

Historical note

Notes and answers

10. Continued fractions

Irrational square roots

Convergence

Purely periodic continued fractions

Pell’s equation

Lagrange’s theorem on quadratic irrationals

Automorphs of the indefinite form ax^{2} – by^{2}

Summary

Historical note

Notes and answers

11. Approximation of irrationals by rationals

Naive approach

Farey sequences

Hurwitz’ theorem

Liouville’s theorem

Summary

Historical note

Notes and answers

Bibliography

Index