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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2507.12397 |
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| _version_ | 1866916864544735232 |
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| author | Katz, Ethan Pratt, Kyle |
| author_facet | Katz, Ethan Pratt, Kyle |
| contents | We investigate the Lebesgue--Nagell equation \begin{align*}
x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We confirm the conjecture unconditionally for $p\leq 13$, and prove the conjecture holds for $p>911$ through a careful application of lower bounds for linear forms in two logarithms. We also show that any ``nontrivial'' solution must satisfy $y > 10^{1000}$. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Lebesgue-Nagell equation $x^2-2 = y^p$ Katz, Ethan Pratt, Kyle Number Theory We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We confirm the conjecture unconditionally for $p\leq 13$, and prove the conjecture holds for $p>911$ through a careful application of lower bounds for linear forms in two logarithms. We also show that any ``nontrivial'' solution must satisfy $y > 10^{1000}$. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature. |
| title | On the Lebesgue-Nagell equation $x^2-2 = y^p$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.12397 |