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Power of a matrix with integer entries?

Source: 38th Brazilian Undergrad MO (2016) - Second Day, Problem 4

November 25, 2016
Brazilian Undergrad MO 2016linear algebramatrix

Problem Statement

Let
A=(</br>4−525−3</br>)A=\left( \begin{array}{cc}</br>4 & -\sqrt{5} \\ 2\sqrt{5} & -3</br>\end{array} \right)
Find all pairs of integers m,nm,n with n≥1n \geq 1 and ∣m∣≤n|m| \leq n such as all entries of An−(m+n2)AA^n-(m+n^2)A are integer.
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