Consider the Fibonacci sequence $\text{fibo}(n)$ and the fractions
$$ A(n) = \frac{\text{fibo}\left(n^2\right)}{\text{fibo}(n)^2} = \frac{b(n)}{c(n)},$$
where the fractions $\frac{b(n)}{c(n)}$ are in reduced form.
Now, it appears that for all $n > 4$:
If $c(n) = 1$, then $b(n)$ is of the form $2^q p$ where $p$ is a prime.
If $c(n) > 1$, then either $c(n) $ or $b(n)$ is of the form $2^q p$ where $p$ is a prime.
So this formula generates a prime for every $n$.
A weaker conjecture is that this holds for the cases when $n$ is a prime.
Are these conjectures true? How can we prove them?