  We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are
integers and $m$ is positive have no nontrivial periodic integral orbits for
$m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely
many values of $k$ but no higher period orbits. We also show that these
statements are true for all quadratic recursions.
