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

TL;DR

This paper investigates the Lebesgue--Nagell equation for odd prime exponents, confirming the conjecture that only trivial solutions exist for primes up to 13 and beyond 911, with nontrivial solutions being extremely large.

Contribution

It unconditionally proves the conjecture for primes up to 13 and for primes greater than 911, and provides bounds on potential nontrivial solutions, advancing understanding of this classical equation.

Findings

Confirmed the conjecture for p ≤ 13

Proved the conjecture for p > 911

Nontrivial solutions must have y > 10^{1000}

Abstract

We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} 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.