Jessica Alessandri

Integral solutions of a Fermat's near miss

In this talk I will consider a "Fermat's near miss", specifically the Diophantine equation x^4+y^4 -8=z^2. Studying the existence and abundance of integral solutions of this kind of equations can be translated in studying the integral points on a del Pezzo surface of degree 2. Rational points on these surfaces have been widely studied, but results on integral points are still missing. I will present a joint work (in progress) with Dan Loughran, where we show that on this surface we have "many" (i.e. a dense set of) integral points.