Interpolation determinants and the Lebesgue--Nagell equation $a^2-D=b^p$

Abstract

We prove the long-standing conjecture that, for every odd prime $p$, the only integral solutions of $a^2-2=b^p$ are $(a,b)=(\pm 1, -1)$. We also completely solve the analogous equations $y^2-D=x^p$ for $D=3$ and $D=5$ and describe a general approach for other positive squarefree values of $D \not \equiv 1 \pmod 8$. The proof refines the interpolation determinant method for linear forms in two logarithms in special cases, introducing new ideas for both the arithmetic lower bounds and the analytic upper bounds.