| Topology
of Numbers |
The idea is for this to be 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. The title of the book will probably be changed to something more informative before it is published, which is likely to be in 2020 or 2021.
You can download a pdf file of the current version of the book, about 275 pages, first posted in September 2019. The main changes from earlier versions occur in the later chapters which have been revised and considerably expanded. There are still a few small topics I would like to add, but apart from this I hope the book is now nearly complete as far as what it covers is concerned. (However, more exercises are needed, especially for the later chapters.)
Chapter 0. Preview
Pythagorean Triples. Rational Points on 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. The Upper Half-Plane Farey Diagram. Relation with Pythagorean Triples.
Chapter 2. Continued Fractions
The Euclidean Algorithm. Determinants and the Farey Diagram. The Diophantine Equation ax+by=n. Infinite Continued Fractions.
Chapter 3. Linear Fractional Transformations
Symmetries of the Farey Diagram. 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
The Four Types of Forms. Equivalence of Forms. The Class Number. Symmetries. 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. The Class Group for Quadratic Forms
Multiplication of Forms. The Class Group for Forms. Finite Abelian Groups. Symmetry and the Class Group. Genus and the Class Group.
Chapter 8. Quadratic Fields
Prime Factorization. Unique Factorization via the Euclidean Algorithm. The Correspondences Between Forms and Ideals. The Ideal Class Group. Unique Factorization of Ideals. Applications to Forms.