On the equation x2 + y3 = z7

This equation is a close cousin of the Fermat equation xn + yn = zn. As for the Fermat equation, one is interested in determining all primitive nontrivial integral solutions. Here nontrivial means that none of the variables vanishes and primitive means that x, y and z have no common prime divisor.

By a general result on generalized Fermat equations, one knows that there exist only finitely many of these solutions. Five such solutions are known; some of them are pretty large, for example

15 312 2832 + 9 2623 = 1137.
The ultimate goal of the project I will be talking about (undertaken jointly with Bjorn Poonen (UC Berkeley) and Ed Schaefer (University of Santa Clara)) is to prove that there are no more than these five solutions. In the talk I will explain our approach to this problem, which involves reducing it to determining the set of rational points on some curves of genus 3.

Michael Stoll