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Are there any books that are substantially based on a geometric approach to explain topics in number theory (elementary and more advanced)?
If so, is such approach -- judging from your teaching (or studying) experience -- fruitful and should it be encouraged in students?
This is likely not exactly what you seek, but it might be interesting nonetheless:
The Geometry of Numbers, C. D. Olds, Anneli Lax, Giuliana P. Davidoff, 2000. (AMS link).
This is a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice points on lines, circles and inside simple polygons in the plane. A minimum of mathematical expertise is required beyond an acquaintance with elementary geometry. The authors gradually lead up to the theorems of Minkowski and others who succeeded him. On the way the reader will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres.
I have not taught number theory, but I do keep an eye out for books with a geometric or visual approach. One that I've come across is Allen Hatcher's draft text Topology of Numbers, available from http://www.math.cornell.edu/~hatcher/TN/TNpage.html. Something to note is that by topology, the author means "the spatial arrangement and interlinking of the components of a system" in this book. Perhaps it's what you're looking for.
(Allen Hatcher is also the author of Algebraic Topology, in which topology is used in its traditional mathematical sense.)
This book when it's finally done might be exactly what you're looking for. I find the sample pages pretty compelling.
Edit: The book has since been published. It's An Illustrated Theory of Numbers by Weissman.
A recent book is Number Theory and Geometry: An Introduction to Arithmetic Geometry by Álvaro Lozano-Robledo, AMS 2019. It's particularly nice for students who want to see how number theory works in the context of curves, especially lines, conics and cubics. It might attract some to studying elliptic curves and arithmetic/diophantine geometry. The author has also written a more advanced, but still introductory book on elliptic curves, modular forms and L-functions.
See https://www.maa.org/press/maa-reviews/number-theory-and-geometry-an-introduction-to-arithmetic-geometry for a review by Abbey Bourdon. Here is a short excerpt:
In Number Theory and Geometry: An Introduction to Arithmetic Geometry, standard topics are covered with the unifying goal of finding rational (or, at times, integral) points on curves. These kinds of problems can be traced back to the Greek mathematician Diophantus in his series of books called Arithmetica, and they have continued to inspire an active area of research today.
