Question 44
The sequence $\langle a_n \rangle$ is defined by the recursion
$$
a_{n+1} = 5a_n^6 + 3a_{n-1}^3 + a_{n-2}^2 \quad \text{for } n \ge 2
$$
and the set of starting values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$.
(i.e., the starting values are these three numbers in arbitrary order.)
Show that the sequence does not contain any sixth power of an integer.
$$
a_{n+1} = 5a_n^6 + 3a_{n-1}^3 + a_{n-2}^2 \quad \text{for } n \ge 2
$$
and the set of starting values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$.
(i.e., the starting values are these three numbers in arbitrary order.)
Show that the sequence does not contain any sixth power of an integer.