| 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 a preliminary draft, so it is incomplete and lacking in polish at certain points.
You can download a pdf file of what currently exists of the book, about 200 pages. This version was posted in January 2019. The main changes from earlier versions occur in Chapters 5-7 which have been revised and expanded. Chapter 7 is still lacking about 10 or 20 pages which I plan to write in the next couple months .
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. The Class Number. Charting All Forms.
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
Prime Factorization. Unique Factorization via the Euclidean Algorithm. The Correspondences Between Forms and Ideals. The Class Group. Unique Factorization of Ideals. [Not yet included: Applications to Forms.]