A Pythagorean Introduction to Number Theory

Abstract

This book came out of an attempt to explain to a class of motivated students at the University of Illinois at Chicago what sorts of problems I thought about in my research. In the course, we had just talked about the integral solutions to the Pythagorean Equation and it seemed only natural to use the Pythagorean Equation as the context to motivate the answer. Basically, I motivated my own research, the study of rational points of bounded height on algebraic varieties, by posing the following question: What can you say about the number of right triangles with integral sides whose hypotenuses are bounded by a large number X? How does this number depend on X? In attempting to give a truly elementary explanation of the solution, I ended up having to introduce a fair bit of number theory, the Gauss circle problem, the Möbius function, partial summation, and other topics. These topics formed the material in Chapter 13 of the present text. Mathematicians never develop theories in the abstract. Despite the impression given by textbooks, mathematics is a messy subject, driven by concrete problems that are unruly. Theories never present themselves in little bite-size packages with bowties on top. Theories are the afterthought. In most textbooks, theories are presented in beautiful well-defined forms, and there is in most cases no motivation to justify the development of the theory in the particular way and what example or application that is given is to a large extent artificial and just “too perfect.” Perhaps students are more aware of this fact than what professional mathematicians tend to give them credit for—and in fact, in the case of the class I was teaching, even though the material of Chapter 13 was fairly technical, my students responded quite well to the lectures and followed the technical details enthusiastically. Apparently, a bit of motivation helps.