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arXiv · 2511.17456

Fermat near misses and the integral Hilbert Property

Jessica Alessandrì·Daniel Loughran
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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.

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Jessica Alessandrì, Daniel Loughran. 2025-11-21. Fermat near misses and the integral Hilbert Property. https://arxiv.org/abs/2511.17456

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