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 350 pages, posted in March 2022, which should be very close to the published version. Publication should be in 2022.

**Chapter 0. Preview**

**Chapter 1. The Farey Diagram**

The Mediant Rule.

Farey Series.

**Chapter 2. Continued Fractions**

Finite Continued Fractions.

Infinite Continued Fractions.

Linear Diophantine Equations

**Chapter 3. Symmetries of the Farey Diagram**

Linear Fractional Transformations.

Translations and Glide Reflections.

**Chapter 4. Quadratic Forms**

The Topograph.

Periodicity.

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.

Representations 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 Rational Equivalence.

**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.