Edward Barbeau - Bringing history close to home: Pell's equation

An excellent topic for acquainting students with the historical flow of mathematical discovery is Pell's equation. For given integer parameters d and k, one seeks integer solutions for x² - dy² = k. Natural questions about numbers, which students can find and investigate themselves, lead to this equation. They can match wits with early Indian and European mathematicians in tackling them. We can trace the evolution from ad hoc (albeit ingenious) to systematic techniques, and follow the progress to a theoretical framework for the mathematical structure of the set of solutions. This is an area that can be handled at many levels of sophistication.