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arXiv · 2507.12397

On the Lebesgue-Nagell equation $x^2-2 = y^p$

Ethan Katz·Kyle Pratt
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Abstract

We investigate the Lebesgue--Nagell equation $$x^2-2=y^p$$ 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.

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Ethan Katz, Kyle Pratt. 2025-07-16. On the Lebesgue-Nagell equation $x^2-2 = y^p$. https://arxiv.org/abs/2507.12397

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