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 title "Topology of Numbers" is intended to convey this idea of a more geometric slant, where we are using the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning of a set with certain specified subsets called open sets. A fair portion of the book is devoted to studying Conway's topographs associated to quadratic forms in two variables, so perhaps the title could have been "Topography of Numbers" instead.

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 160 pages. This version was posted in July 2017. The main changes from the previous version occur in Chapter 6 which has been considerably expanded.


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. Representing Primes. Genus and Characters. Representing Non-primes. Proof of Quadratic Reciprocity.

Chapter 7. Quadratic Fields

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