Hatcher A. Topology of Numbers

Cornell University, 2019. — 213 p.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.
Prerequisites for reading the book are fairly minimal, hardly going beyond high school mathematics for the most part. 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 congruence theory as a separate topic and will instead just use congruences 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. Among these is quadratic reciprocity, where we give Eisenstein’s classical proof since it involves some geometry. This version was posted in January 2019. The main changes from earlier versions occur in Chapters 5-7 which have been revised and expanded.Preview.
The Farey Diagram.
Continued Fractions.
Linear Fractional Transformations.
Quadratic Forms.
Classification of Quadratic Forms.
Representations by Quadratic Forms.
Quadratic Fields.