$31 \cdot 37$ exists an integer $n$, such that $\sqrt{p+n}+\sqrt{n}$ is an integer for the prime $p$
(A) does not exist.
(B) there is only one.
(C) more than one, but a finite number.
(D) there are infinitely many.
(China High School Mathematics League, 1996)
[Solution] Let $p$ be any odd prime, and $p=2k+1$. Then let $n=k^2$, we have
$$
\sqrt{p+n}+\sqrt{n}=\sqrt{2k+1+k^2}+\sqrt{k^2}=2k+1 \text{.}
$$

Therefore, every odd prime has the property mentioned in the problem.
Hence, the answer is $(D)$.