Question 149
Let $c$ be a positive integer. The sequence $a_1, a_2, \dots, a_n, \dots$ is defined by $a_1 = c$, and $a_{n+1} = a_n^2 + a_n + c^3$, for every positive integer $n$. Find all values of $c$ for which there exist some integers $k \ge 1$ and $m \ge 2$, such that $a_k^2 + c^3$ is the $m$th power of some positive integer.