For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.
Classical and modular approaches to exponential Diophantine equations
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On the Lebesgue-Nagell equation $x^2-2 = y^p$
For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.