Proving a statement about prime numbers

Question

Let $p_1,p_2,p_3,\cdots$ be all the primes sorted in an increasing order.

Is $p_1p_2p_3\cdots p_i + 1$ is always prime? Why? How can I prove that?

Answer 1

$P_n=p_1p_2\cdots p_n+1$ can't be divisible by any of $p_1,p_2,\ldots,p_n$, and so $P_n$ must be prime if $P_n<p_{n+1}^2.$ So in particular 3 is prime since $3<3^2$, 7 is prime since $7<5^2,$ and 31 is prime since $31<7^2$. But the argument doesn't work for larger values, and in particular it fails for $13\#+1$, $17\#+1$, $19\#+1$, etc.