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

The periodic integral orbits of polynomial recursions with integer coefficients