I've seen a problem that gives $$A= \begin{bmatrix} 4 & -\sqrt5\\ 2\sqrt5 & -3 \end{bmatrix}$$
and asks to find all pairs $(n,m) \in \mathbb{N} \times \mathbb{Z}$, with $|m|\leq n$ such that:
$A^n -(n^2+m)A$
has all integer entries. I've calculated the char. polynomyal (which is $p(\lambda) = \lambda^2-\lambda-2$) but I have no idea of how can I use it to obtain what the problem asks. Also, probably is it better to calculate for small exponents n but I don't see an imediate reason for that.
