Hatcher Allen. Topology of Numbers

2017. — 147 p.This book provides an introduction to Number Theory from a somewhat unusual geometric point of view. It might have been called “Geometry of Numbers" if this phrase did not already have a well-established meaning rather different from what we have in mind here. Instead we have chosen the title Topology of Numbers where we are using the term “Topology" with its general meaning of “the spatial arrangement and interlinking of the components of a system" rather than its standard mathematical meaning involving open sets, etc.
The principal geometric theme is the so-called Farey diagram which dates back to an 1894 paper of Adolf Hurwitz. This is a two-dimensional figure which displays certain relationships between rational numbers beyond just their usual distribution along the one-dimensional real number line. Among the things the diagram elucidates are Pythagorean triples, the Euclidean algorithm, Pell’s equation, continued fractions, Farey sequences (of course!), two-by-two matrices with integer entries, and quadratic forms in two variables with integer coefficients, where this last case uses John Conway’s marvelous idea of the topograph of such a form. A good part of the book is devoted to this last topic, and in fact an alternative title for the book might be “The Topography of Numbers".
Prerequisites for reading the book are fairly minimal, hardly going beyond high school mathematics. One topic that often forms a significant part of elementary number theory courses is congruences modulo an integer n. It would be helpful if the reader has already seen and used these a little, but we will not develop them systematically and instead just use them as the need arises, proving whatever nontrivial facts are required including several of the basic ones that form part of a standard introductory number theory course. This includes quadratic reciprocity, where we give Eisenstein’s classical proof since it involves some geometry.A Preview.
The Farey Diagram.
Continued Fractions.
Linear Fractional Transformations.
Quadratic Forms.
Classification of Quadratic Forms.
Representations by Quadratic Forms.
Quadratic Fields.