Fermat near misses and the integral Hilbert Property

2025/11/21 by Jessica Alessandrì, Alessandrì, Jessica, Daniel Loughran +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2511.17456

Abstract

We consider the Diophantine equation x⁴ + y⁴ - w² = n for n ∈ 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.

Citations

• Hilbert properties of varieties
• The Mordell-Schinzel conjecture for cubic diophantine equations
• Cubic surfaces with infinite, discrete automorphism group
• An Hilbert Irreducibility Theorem for integral points of del Pezzo surfaces
• A note on Fermat's Last Theorem for n=4
• Rank jumps on elliptic surfaces and the Hilbert property
• Cracking the problem with 33
• Hilbert property for double conic bundles and del Pezzo varieties
• Elliptic Fibrations and Hilbert Property
• Non-rational varieties with the Hilbert Property
• On the Hilbert Property and the Fundamental Group of Algebraic Varieties
• Geometry and arithmetic of certain log K3 surfaces
• A generalization of Dirichlet's unit theorem
• On the arithmetic of del Pezzo surfaces of degree 2
• Density of integral points on algebraic varieties

Cited by

• Hilbert properties of varieties
• Hilbert property for low-genus families and degree-one del Pezzo surfaces

Related

• A note on Fermat's Last Theorem for n=4
• Diophantine Equation n(x⁴+y⁴)=z⁴+w⁴
• A Quartic Identity Related to Fermat-Type Equations
• An Hilbert Irreducibility Theorem for integral points of del Pezzo surfaces
• Solution of certain Diophantine equations in Gaussian integers