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 $D\not\equiv 1(\mathrm{mod}8)$. 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.

arXiv comment44 pages, comments very welcome!