Fermat near misses and the integral Hilbert Property

TL;DR

This paper investigates the solutions to a specific quartic Diophantine equation related to Fermat's Last Theorem, using algebraic geometry to demonstrate the infinitude or non-thinness of solutions for certain parameters.

Contribution

It introduces a geometric approach via del Pezzo surfaces to analyze the solution set of a Fermat-related Diophantine equation, establishing non-thinness results.

Findings

Infinite solutions for certain n values.

Solutions form a non-thin set.

Method applies to double conic bundle surfaces.

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.