Ramanujan-Nagell cubics

Source abstract

A well-known result of Beukers on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity $|x^2-2^n|$. In this paper, we derive an inequality of the shape $|x^3-2^n| \geq x^{4/3}$, valid provided $x^3 \neq 2^n$ and $(x,n) \neq (5,7)$, and then discuss its implications for a variety of Diophantine problems.