Rational periodic points of x^d+ c and Fermat-Catalan equations

Abstract

In this paper, we study rational periodic points of polynomial fd,c(x) = xd + c over the field of rational numbers, where d is an integer greater than two. For period two, we describe periodic points for degrees d = 4, 6. We also demonstrate the nonexistence of rational periodic points of exact period two for d = 2k such that 3|2k - 1 and k has a prime factor greater than three. Moreover, assuming the abc-conjecture, we prove that fd,c has no rational periodic point of exact period greater than one for sufficiently large integer d and c ≠ - 1.