We investigate the Lebesgue--Nagell equation align* x²-2=yᵖ 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 > 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.
Katz et al. (Wed,) studied this question.