Rational Periodic Points of xd + c and Fermat-Catalan Equations

Abstract

We study rational periodic points of polynomial fd,c(x) = x d + c over the field of rational numbers, where d is an integer greater than 2 and c 6= −1. For period 2, we classify all possible periodic points for degrees d = 4, 6, and d = 2k for positive integer k > 3 such that 2k − 1 is divisible by 3. Moreover, assuming the abc-conjecture, we prove that fd,c has no rational periodic point of exact period greater than 1 for sufficiently large integer d.