Fermat near misses and the integral Hilbert Property
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911315950305280 |
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| author | Alessandrì, Jessica Loughran, Daniel |
| author_facet | Alessandrì, Jessica Loughran, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fermat near misses and the integral Hilbert Property Alessandrì, Jessica Loughran, Daniel Number Theory Algebraic Geometry 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. |
| title | Fermat near misses and the integral Hilbert Property |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2511.17456 |