The Periodic Integer Orbits of Polynomial Recursions With Integer Coefficients - Sedaghat, Hassan (Sarajevo Journal of Mathematics, 2024)
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≥3. 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.