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We study rational periodic points of polynomial $f_{d,c}(x)=x^d+c$ over thefield of rational numbers, where $d$ is an integer greater than 2. For period2, we classify all possible periodic points for degrees $d=4,6$. We alsodemonstrate 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 norational periodic point of exact period greater than 1 for sufficiently largeinteger $d$ and $c\neq -1$.

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