Topology of Numbers

This is an introduction to elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. A large part of the book is devoted to studying quadratic forms in two variables with integer coefficients, a very classical topic going back to Fermat, Euler, Lagrange, Legendre, and Gauss, but from a perspective that emphasizes Conway's much more recent notion of the topograph of a quadratic form.

You can download a pdf file of the current version of the book, around 300 pages, posted in October 2020. The book is now essentially complete and I am trying to do finial polishing. A few more exercises need to be added, especially in the later chapters. Publication should be in early 2021.


Table of Contents

Chapter 0. Preview

Chapter 1. The Farey Diagram

The Mediant Rule.

Farey Series.

Chapter 2. Continued Fractions

The Euclidean Algorithm.

Linear Diophantine Equations

Infinite Continued Fractions.

Chapter 3. Symmetries of the Farey Diagram

Linear Fractional Transformations.

Continued Fractions Again.

Chapter 4. Quadratic Forms

The Topograph.

Periodic Separator Lines.

Continued Fractions Once More.

Pell's Equation.

Chapter 5. Classification of Quadratic Forms

The Four Types of Forms.

Equivalence of Forms.

The Class Number.

Symmetries of Forms.

Charting All Forms.

Chapter 6. Representations by Quadratic Forms

Three Levels of Complexity.

Representions in a Fixed Discriminant.

Genus and Characters.

Proof of Quadratic Reciprocity.

Chapter 7. The Class Group for Quadratic Forms

Multiplication of Forms.

The Class Group for Forms.

Finite Abelian Groups.

Symmetry and the Class Group.

Genus and the Class Group.

Chapter 8. Quadratic Fields

Prime Factorization.

Unique Factorization via the Euclidean Algorithm.

The Correspondence Between Forms and Ideals.

The Ideal Class Group.

Unique Factorization of Ideals.

Applications to Forms.