The periodic integral orbits of polynomial recursions with integer coefficients

TL;DR

This paper investigates the existence of periodic integral orbits in polynomial recursions with integer coefficients, establishing nonexistence results for higher periods and specific cases for quadratic recursions.

Contribution

It proves that polynomial recursions of degree greater than two have no nontrivial periodic integral orbits, and characterizes the behavior of quadratic recursions regarding cycles.

Findings

No nontrivial periodic orbits for $m \,\geq\, 3$ in polynomial recursions.

Existence of integral two-cycles for quadratic recursions with infinitely many $k$.

Higher period orbits do not exist for quadratic recursions.

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 $m\geq3$. 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.