  We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq
\exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's
explicit formula for the Riemann zeta-function with explicit bounds on the
error term. We use this along with other recent explicit estimates regarding
the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we
show that there is a prime between any two consecutive $m$th powers for $m \geq
4.971 \times 10^9$.
