3. There exists an integer $n$ such that $\sqrt{p+n}+\sqrt{n}$ is an integer for the prime number $p$
A. None
B. Only one
C. More than one, but a finite number
D. Infinitely many

The prime number $p$ for which there exists an integer $n$ such that $\sqrt{p+n}+\sqrt{n}$ is an integer.
According to the problem, $n$ is a perfect square. Let $p=2k+1$, and take $n=k^{2} \Rightarrow \sqrt{p+n}+\sqrt{n}=2k+1=p$. Therefore, there are infinitely many prime numbers $p$ that satisfy the condition. So the answer is $D$.

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