$\mathbb{Q}$-curves and the Lebesgue-Nagell equation

Michael A. Bennett, Philippe Michaud-Jacobs, Samir Siksek

In this paper, we consider the equation \[ x^2 - q^{2k+1} = y^n, \qquad q \nmid x, \quad 2 \mid y, \] for integers $x,q,k,y$ and $n$, with $k\ge 0$ and $n\ge 3$. We extend work…