Rational periodic points of xd + c and Fermat–Catalan equations

Abstract

In this paper, we study rational periodic points of polynomial [Formula: see text] over the field of rational numbers, where [Formula: see text] is an integer greater than two. For period two, we describe periodic points for degrees [Formula: see text]. We also demonstrate the nonexistence of rational periodic points of exact period two for [Formula: see text] such that [Formula: see text] and [Formula: see text] has a prime factor greater than three. Moreover, assuming the [Formula: see text]-conjecture, we prove that [Formula: see text] has no rational periodic point of exact period greater than one for sufficiently large integer [Formula: see text] and [Formula: see text].