On the equation $y^2=x^6+k$

Abstract

We find all rational solutions to the equation $y^2=x^6+k$, where $k=-39$ or $k=-47$. These are the two unsolved cases for $k$ in the range $\left|k\right|\le 50$. Our method is elementary and can be applied to other situations to find all rational points on certain curves of genus $2$.