MathNet 2015-part-04 — 191 problems (EN–VI) Question 147

Determine all natural numbers $k \geq 1$ and $n \geq 2$ for which there exist $A, B \in \mathcal{M}_{n}(\mathbb{Z})$ such that $A^{3}=O_{n}$ and $A^{k} B+B A=I_{n}$.