Fermat near misses and the integral Hilbert Property

Abstract

We consider the Diophantine equation $x^4+y^4-w^2=n$ for $n\in \mathbb{Z}$, which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we show that the set of solutions is infinite, or more generally not thin. Our approach is via the geometry of del Pezzo surfaces of degree $2$, and we prove a more general result on non-thinness of integral points on double conic bundle surfaces.

24 pages. Minor changes to introduction; improved Example 4.6 and proof of Lemma 5.4