  Mordell equations are celebrated equations within number theory and are named
after Louis Mordell, an American-born British mathematician, known for his
pioneering research in number theory. In this paper, we discover all Mordell
equations of the form $y^2 = x^3 + k$, where $k \in \mathbb Z$, with exactly
$|k|$ integral solutions. We also discover explicit bounds for Mordell
equations, parameterized families of elliptic curves and twists on elliptic
curves. Using the connection between Mordell curves and binary cubic forms, we
improve the lower bound for the number of integral solutions of a Mordell curve
by looking at a pair of curves with unusually high rank.
