Let P denote the product of first n prime numbers (with n > 2). Find all values of $n$ .

Question

Let P denote the product of first n prime numbers (with $n > 2$). For what values of n we have.
$1.$ $P - 1$ is a perfect square.
$2.$ $P + 1$ is a perfect square.

What I Tried: I have almost solved Part $2$, and I have some ideas of Part $1$.
For Part $2$, let $P + 1 = k^2$ .
$\rightarrow P = (k + 1)(k - 1) . $

So, after experimenting with some values of $n$, I observed that $P + 1$ is a perfect square when $n = 2,4$ .
When it is $3*5$ it works, and in the next $n$ it works as we get $3 * 5 * 7 * 11 = 33 * 35$ .
I was not able to get any other $n$, and if there is a proof, I need help to get it.

For Part $1$, I didn't know how to do it algebraically, so I experimented with some $n$. Obviously $P - 1$ won't be prime for $n = 2,4$. However, I noticed that Part $1$ does not work for any $n$, but I was not able to get a proof for it.

Can someone help me with these two parts? Thank You.

Edit:- I just now noticed that I thought myself that $n$ meant the starting prime value, which is wrong. It meant that $n$ is the total no. of starting primes, so my approach for Part $2$ was wrong.

Answer 1

Hints :

For $n>2$,

• $(P-1)$ is $2$ modulo $3$.
• $(P+1)$ is $3$ modulo $4$.