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

Abstract

We investigate the Lebesgue--Nagell equation $$x^2-2=y^p$$ in integers $x,y,p$ with $p\ge 3$ 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 $p\le 13$, 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.

46 pages. Published in Ramanujan J. 69 (2026), no. 3, Paper No. 75, 56 pp