This is an introduction to elementary number theory from a geometric point of view, in contrast 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 book has been published by the AMS in 2022 as a paperback, ISBN 978-1-4704-5611-5. See the AMS webpage for the book.
You can download a pdf file of the book, around 350 pages. This is a minor revision of the print version which you can find here if you would like to see that for some reason.
Please send any corrections that you find to me at the email address listed on my homepage.
Table of Contents
- Chapter 0. A Preview
- Chapter 1. The Farey Diagram
- 1.1 The Mediant Rule. 1.2 Farey Series.
- Chapter 2. Continued Fractions
- 2.1 Finite Continued Fractions. 2.2 Infinite Continued Fractions. 2.3 Linear Diophantine Equations.
- Chapter 3. Symmetries of the Farey Diagram
- 3.1 Linear Fractional Transformations. 3.2 Translations and Glide Reflections.
- Chapter 4. Quadratic Forms
- 4.1 The Topograph. 4.2 Periodicity. 4.3 Pell's Equation.
- Chapter 5. Classification of Quadratic Forms
- 5.1 The Four Types of Forms. 5.2 Equivalence of Forms. 5.3 The Class Number. 5.4 Symmetries of Forms. 5.5 Charting All Forms.
- Chapter 6. Representations by Quadratic Forms
- 6.1 Three Levels of Complexity. 6.2 Representations in a Fixed Discriminant. 6.3 Genus and Characters. 6.4 Proof of Quadratic Reciprocity.
- Chapter 7. The Class Group for Quadratic Forms
- 7.1 Multiplication of Forms. 7.2 The Class Group for Forms. 7.3 Finite Abelian Groups. 7.4 Symmetry and the Class Group. 7.5 Genus and the Class Group. 7.6 Rational Equivalence and Rational Forms.
- Chapter 8. Quadratic Fields
- 8.1 Prime Factorization. 8.2 Unique Factorization via the Euclidean Algorithm. 8.3 The Correspondence Between Forms and Ideals. 8.4 The Ideal Class Group. 8.5 Unique Factorization of Ideals. 8.6 Applications to Forms.