**On the equation ***x*^{2} + *y*^{3}
= *z*^{7}

This equation is a close cousin of the Fermat equation
*x*^{n} + *y*^{n}
= *z*^{n}. 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

15 312 283^{2} + 9 262^{3} = 113^{7}.
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*