On the Lebesgue-Nagell equation x²-2 = yᵖ

We investigate the Lebesgue--Nagell equation {align*} x^2-2=y^p {align*} in integers $x,y,p$ with p≥ 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≤ 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 > 10¹⁰⁰⁰. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature.