ISL 2012 N4

An integer $a$ is called friendly if the equation $\left(m^{2}+n\right)\left(n^{2}+m\right)=a(m-n)^{3}$ has a solution over the positive integers. a) Prove that there are at least 500 friendly integers in the set $\{1,2, \ldots, 2012\}$. b) Decide whether $a=2$ is friendly.