CN-LEAGUE 1996 Problem 3

For a prime number $p$ , there exists $n\in\mathbb{Z}_+$ , $\sqrt{p+n}+\sqrt{n}$ is an integer, then $\text{(A)}$ there is no such $p$ $\text{(B)}$ there in only one such $p$ $\text{(C)}$ there is more than one such $p$ , but finitely many $\text{(D)}$ there are infinitely many such $p$