Computing integral points on Mordell's elliptic curves

We use Mordell's elliptic curves $E_k$ (see below) to illustrate our algorithm for computing all integral points on \textit{any} given elliptic curve over the rationals (see [5]) and apply it to determine the integral points on $E_k$ for $k$ within the range $\vert k\vert \leq 10, 000$. Actually, the calculations can be extended to $\vert k\vert\leq 100, 000$. In this larger range Hall's conjecture holds with $c_\epsilon = 5$.