In post I would like to ask about an example of a diophantine equation having the form
$$P(x)=p_n\#\tag{1}$$ where $P(X)\in\mathbb{Z}[X]$ is a polynomial of degree $\deg(P)=2$ and $p_n\#$ denotes the primorial of order $n$, that is $$p_n\#=\prod_{k=1}^n p_k,$$ being $p_k$ the $k$th prime number. This is the Wikipedia's article dedicated to Primorial.
Equations of the form $(1)$ were in the literature for different aritmetic functions in the corresponding RHS.
Our problem. We consider the solutions $(n,x)$, where $n\geq 1$ is an integer and also $x\geq 1$ is a positive integer, of $$\frac{x(x+1)}{2}=p_n\#.\tag{2}$$ One finds easily two solutions by observation $(n,x)=(2,3)$ and $(n,x)=(4,20)$.
Question. Can be proven that our problem has finilety many solutions $(n,x)$? Many thanks.
Remark 1 (see the comments). Thanks to the users in the comments, now the question can be rephrased to (see also the references to OEIS)
Is there $n>4$ such that $8(p_n\#)+1$ is a square number?
I am waiting thus what work can be done to prove that the equation has finitely many solutions $(n,x)$, or at least if some user can to find a new solution (after that there are some answers I think that it is reasonable that I should to accept an answer.)
Remark 2. Using Sage Cell Server (choose GP as Language) with a Pari/GP program, this (and similar) code
for (n = 1, 200,for (x = 1, 10000, if (x*(x+1)/2==prod( k=1, n, prime(k)), print(n))))
I cann't find more solutions. I have tried some reasonings (splitting the primorial in two factors or use the size of primorials) but these do not work.