Rational Periodic Points of $x^d+c$ and Fermat-Catalan Equations

TL;DR

This paper investigates rational periodic points of polynomials of the form x^d + c over the rationals, classifies period 2 points for specific degrees, and, assuming the abc-conjecture, shows the absence of higher period points for large degrees.

Contribution

It classifies all period 2 rational points for degrees 4 and 6, and proves, under the abc-conjecture, that no rational periodic points of period greater than 1 exist for large degrees.

Findings

Classified all period 2 points for degrees 4 and 6.

Proved nonexistence of period 2 points for certain even degrees.

Under abc-conjecture, no rational periodic points of period > 1 for large degrees.

Abstract

We study rational periodic points of polynomial $f_{d,c}(x)=x^d+c$ over the field of rational numbers, where $d$ is an integer greater than 2. For period 2, we classify all possible periodic points for degrees $d=4,6$. We also demonstrate the nonexistence of rational periodic points of exact period 2 for $d=2k$ such that $3\mid 2k-1$ and $k$ has a prime factor greater than 3. Moreover, assuming the $abc$-conjecture, we prove that $f_{d,c}$ has no rational periodic point of exact period greater than 1 for sufficiently large integer $d$ and $c\neq -1$.