Studying when $P_n=(p_1\cdot p_2\cdots p_n)+1$ is a square number

Question

Let $p_n$ be the $n$th prime number. I need to find under which conditions the number $$P_n=(p_1\cdot p_2\cdots p_n)+1$$ is a square number. So far I have seen that $$P_1 = 2+1 =3$$ $$P_2 = 2\cdot3+1 =7$$ $$P_3 = 2\cdot3\cdot5+1 =31$$ give all prime numbers, this is that $P_n$ is always a prime number, and so it can never be a square number. But I can not find the exact way to prove it. I though of trying something like the prove for the Euclidean Theorem that states that there are infinite prime numbers but I can not figure it out. Thank you.

Answer 1

Hint:

An odd square number has the form $4n+1$ but $(p_1\cdot p_2\cdots p_n)+1=2d+1$ where $d$ is odd.