The periodic integral 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 integral orbits for $m\ge 3$. 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.

19 pages, 2 figures