Topology of Numbers

The plan is for this to be an introductory textbook on elementary number theory from a somewhat geometric point of view. It might have been called "Geometry of Numbers" except that this phrase already has an established meaning somewhat different from the circle of ideas here, which centers on the Farey diagram and Conway's topographs of quadratic forms.

The present version of the book is still just a preliminary draft, so it needs much polishing and some sections are still incomplete. Chapters are being posted as the semester progresses.

Chapter 0. A Preview: Pythagorean Triples

Chapter 1: The Farey Diagram

1. Constructing the Farey Diagram

2. Continued Fractions

3. Linear Fractional Transformations

1. Topographs

2. The Classification of Quadratic Forms

3. Representing Numbers by Forms (parts 1 and 2 together)

Chapter 3: Quadratic Fields

Here are the first few pages

The book might be described very loosely as an attempt to make Gauss's Disquisitiones Arithmeticae (1801), which can be considered as the first textbook in number theory, into a modern undergraduate textbook, focusing on the parts of Gauss's book that lend themselves more to geometric interpretations. Another inspiration for the book is Davenport's classic textbook The Higher Arithmetic (1952).