Question 33
Define sequence of positive integers $\left(a_{n}\right)$ as $a_{1}=a$ and $a_{n+1}=a_{n}^{2}+1$ for $n \geq 1$. Prove that there is no index $n$ for which
$$
\prod_{k=1}^{n}\left(a_{k}^{2}+a_{k}+1\right)
$$
is a perfect square.
$$
\prod_{k=1}^{n}\left(a_{k}^{2}+a_{k}+1\right)
$$
is a perfect square.