Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$is a perfect square.
I am getting the only thing that The product is $\frac{a_{n+1}^2 - a_{n+1} + 1}{a^2 - a + 1}$ but
how to prove it's not a perfect square.
Anyone with any ideas?
you prove that: $(a_k^2+a_k+1,a_l^2+a_l+1)=1$ ( not easy but not difficult , just calc gcd like normal) so that if that product be square it force:
$a_k^2+a_k+1$ is square but it wrong