Science Seminar Series—Finding Simple Solutions

Suppose we manage to find a solution to an equation. Is it the “simplest” solution we could have found? For example, we might note that x = 374 is a solution to x^3 – 587x^2 + 80084x – 157828, but if we think even less hard about it, so is x = 2. Sometimes “small” is what we mean by “simple,” but sometimes we need of notion of “simple” with a little more oomph. (Are 1/7 or sqrt(2)-1 simpler than 1? They’re smaller!) In this talk, we’ll see some of analytic number theory’s tools for finding “simple” solutions to Diophantine equations.