The Periodic Integer Orbits of Polynomial Recursions With Integer Coefficients

Abstract

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 integer orbits for $m\geq3$. If $m=2$ then we show that the recursion has integer two-cycles for infinitely many values of $k$ but no higher period orbits. We further show that these statements are true for all quadratic recursions and comment on possible higher order extensions.