2012 IMO Shortlist 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.

如果方程 $\left(m^{2}+n\right)\left(n^{2}+m\right)=a(m-n)^{3}$ 在正整数上有解，则整数 $a$ 被称为友好的。 a) 证明集合 $\{1,2, \ldots, 2012\}$ 中至少有 500 个友好整数。 b) 判断$a=2$是否友好。