$30 \cdot 44$ For any given natural number $n$, if $n^{6}+3 a$ is the cube of a positive integer, where $a$ is a positive integer, then
(A) There are infinitely many such $a$.
(B) Such $a$ exist, but there are only finitely many.
(C) Such $a$ do not exist.
(D) None of the conclusions (A), (B), (C) are correct.
(China High School Mathematics League, 1987)
[Solution] Take $a=\frac{1}{3}\left[(3 k)^{3}+3(3 k)^{2} n^{2}+3(3 k) n^{4}\right]$, where $k=1,2, \cdots$ Then $a$ is clearly a positive integer, and there are infinitely many, and
$$
n^{6}+3 a=\left(3 k+n^{2}\right)^{3}
$$

is a cube of a positive integer. Therefore, the answer is $(A)$.