Topology of Numbers

The plan is for this to be an introductory textbook on elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. The book might have been called "Geometry of Numbers" except that this already has an established meaning different from what we have in mind here. In choosing the name "Topology of Numbers" we mean the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning. Perhaps the term "Topography" conveys the idea better, and indeed a fair amount of the book is devoted to studying Conway's topographs associated to quadratic forms in two variables.

The current version of the book is still just a preliminary draft, so it is incomplete and lacking in polish at many points.

You can download a pdf file of what currently exists of the book, about 140 pages.

Table of Contents

Chapter 0. Preview

Pythagorean Triples. Rational Points on Other Quadratic Curves. Rational Points on a Sphere. Pythagorean Triples and Quadratic Forms. Pythagorean Triples and Complex Numbers. Diophantine Equations.

Chapter 1. The Farey Diagram

The Diagram. Farey Series. Other Versions of the Diagram. Relation with Pythagorean Triples. The Determinant Rule for Edges.

Chapter 2. Continued Fractions

The Euclidean Algorithm. Connection with the Farey Diagram. The Diophantine Equation ax+by=n. Infinite Continued Fractions.

Chapter 3. Linear Fractional Transformations

Symmetries of the Farey Diagram. Seven Types of Transformations. Specifying Where a Triangle Goes. Continued Fractions Again. Orientations.

Chapter 4. Quadratic Forms

The Topograph. Periodic Separator Lines. Continued Fractions Once More. Pell's Equation.

Chapter 5. Classification of Quadratic Forms

Hyperbolic Forms. Elliptic Forms. Parabolic and 0-Hyperbolic Forms. Equivalence of Forms. Symmetries. Class Number.

Chapter 6. Representations by Quadratic Forms

Three Levels of Complexity. A Criterion for Representability. Proof of Fermat's Theorem on Sums of Two Squares. Primes Represented in a Given Discriminant. Quadratic Reciprocity.

Chapter 7. Quadratic Fields

Primes and Units. The Norm. Prime Factorizations. Unique Factorization via the Euclidean Algorithm. Other Instances of Unique Factorization.