The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty subset $\mathcal I$ of $\{1,\dots,n\}$ and let $P_{\mathcal I}=2\prod_{i\in I}p_i+1$.
Is there an algebraic extension $\Bbb Z[\alpha_1,\dots,\alpha_m]$ of $\Bbb Z$ such that there exists $t$ elements (possibly indistinct) $a_j\in\Bbb Z[\alpha_1,\dots,\alpha_m]$ for $j\in\{1,\dots,t\}$ such that for every $\mathcal I\subseteq\{1,\dots,n\}$ there is a subset $\mathcal J$ from $\{1,\dots,t\}$ such that $P_{\mathcal I}=2\prod_{i\in I}p_i+1=\prod_{j\in\mathcal J}a_j$ holds?
If so how big is least $t$ and $m$ compared to $n$ and how big are norms of $a_j$?
I think $t=O(n)$ and $|a_j|=O(n)$ (where $|a_j|$ is norm of $a_j$) could be possible.