Example 10 There exists a positive integer $n$, such that $\sqrt{p+n}+\sqrt{n}$ is an integer for the prime number $p(\quad)$.
(A) does not exist
(B) there is only one
(C) there are more than one but only a finite number
(D) there are infinitely many
(1996, National High School Mathematics Competition)
Analysis: Let $\sqrt{p+n}+\sqrt{n}=k(k$ be a positive integer), making the problem a prime solution equation for $p$. Rearranging and squaring gives $p=k(k-2 \sqrt{n})$.
Since $p$ is a prime, we can let $k-2 \sqrt{n}=1$, thus,
$$
k=2 \sqrt{n}+1=p \text {. }
$$

Obviously, only the square of a composite number for $n$ will do. Therefore, the answer is (D).