IrMO 2003

find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$

For each positive integer $k$ , let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$ . Calculate $$\sum_{k=1}^{2003} (a_k-b_k).$$