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

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 $c\neq -1$.