I am asking this question exclusively as a reference request for equations similar than those that shows the section D25 of [1], but now involving primorials (see this MathWorld) instead of factorials.
Motivation. Motivated by the notion that diophantine equations are interesting in mathematics, and that there are two unsolved problems in the core of number theory, the Riemann Hypothesis and the abc conjecture, these days I was thinking in diophantine equations involving the radical of an integer $\operatorname{rad}(n)$ or primorials $N_k$. I am curious about if this idea is interesting (create ad hoc equations involving primorials $N_n$ or $\operatorname{rad}(n)$) or was in the literature. This was my motivation to ask this
Question. Were in the literature equations involving primorials (see my idea below this question), explicitly $N_n=\prod_{k=1}^n p_k$ for $n>1$, where thus $p_k$ is the $k$th prime number? Please write the references and I try to find such literature and read some aspects of those. Many thanks.
I am asking for equations likes these, for example
Solve for primorials with $k\geq 1$ and integers $x\geq 1$ $$N_k+1=x^2.\tag{EP1}$$ Solve for primorials with $k\geq 1$ $$N_k=x^a\pm y^a,\tag{EP2}$$ for positive integers satisfying $\gcd(x,y)=1$ and $a>2$ also integer.
Solve for integers $k\geq 1$ and integers $y>1$ $$(x+2)(x+3)(x+5)\cdot\ldots\cdot(x+p_k)=y^2-1.\tag{EP3}$$
I know that the case $k=1$ of a variation of $(EP2)$ $$x^y-y^x=N_k$$ was in the literature as [2] (in spanish).
References:
[1] Richard K. Guy, Unsolved Problems in Number Theory, Volume I, Springer (1994).
[2] Xavier Ros, Problema 137, La Gaceta de la RSME, Vol. 13 (2010), Núm. 3, Pág. 511.