  In this paper, we establish discrete Hardy-Rellich inequalities on
$\mathbb{N}$ with $\Delta^\frac{\ell}{2}$ and optimal constants, for any $\ell
\geq 1$. As far as we are aware, these sharp inequalities are new for $\ell
\geq 3$. Our approach is to use weighted equalities to get some sharp Hardy
inequalities using shifting weights, then to settle the higher order cases by
iteration. We provide also a new Hardy-Leray type inequality on $\mathbb{N}$
with the same constant as the continuous setting. Furthermore, the main ideas
work also for general graphs or the $\ell^p$ setting.
