5. Satisfy
$$
y=\sqrt[3]{x+31}+\sqrt[3]{x+2012}
$$

the integer pairs $(x, y)$ ().
(A) Only one pair
(B) Exactly two pairs
(C) At least three pairs
(D) Do not exist
5. B.

Let $a^{3}=x+31, b^{3}=x+2012(a, b \in \mathbf{Z})$.
Then $\boldsymbol{y}=a+b$,
$$
\begin{array}{l}
b^{3}-a^{3}=(b-a)\left[(b-a)^{2}+3 a b\right] \\
=1981=7 \times 283 .
\end{array}
$$

Since $b>a$, we have
$$
\begin{array}{l}
\left(b-a,(b-a)^{2}+3 a b\right) \\
=(1,1981),(7,283),(283,7),(1981,1) .
\end{array}
$$

Solving, we get $(a, b)=(6,13)$ or $(-13,-6)$.
Thus, $(x, y)=(185,19)$ or $(-2228,-19)$.