We make use of Bonse's inequality. Let $p_u$ denote the $u$th prime and suppose that $p$ is the $n$'th prime. Then for $n \geqslant 5$, from Bonse's inequality we have $p_n + 1 = p_1 p_2 \cdots p_{n-1} > p_{n}^2 $, impossible. So $n \leqslant 4$ and so $p = 2, 3, 5, 7$, of which only $p = 5$ works.