On Sets $X \subseteq {\mathbb N}$ for Which an Explicitly Known Algorithm Returns an Integer \(n\) Such That \(\max(X)<n\) if \(X\) is Finite

Let {\(\Gamma(k)\)} denote {$(k-1)!$}, and let {\(\Gamma_{n}(k)\)} denote {$(k-1)!$}, where {$n \in \{3,\ldots,16\}$} and {\(k \in \{2\} \cup [2^{\textstyle 2^{n-3}}+1,\infty) \cap \mathbb N\)}. For an integer {$n \in \{3,\ldots,16\}$}, let \(\Sigma_n\) denote the following statement: if a system of equations {${\mathcal S} \subseteq \{\Gamma_{n}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$} with \(\Gamma\) instead of \(\Gamma_{n}\) has only finitely many solutions in positive integers {\(x_1,\ldots,x_n\)}, then every tuple {$(x_1,\ldots,x_n) \in (\mathbb N \setminus \{0\})^n$} that solves the original system ${\mathcal S}$ satisfies {\(x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}\)}. Our hypothesis claims that the statements {\(\Sigma_{3},\ldots,\Sigma_{16}\)} are true. The statement {\(\Sigma_6\)} proves the following implication: if the equation {\(x(x+1)=y!\)} has only finitely many solutions in positive integers \(x\) and \(y\), then each such solution {$(x,y)$} belongs to the set {$\{(1,2),(2,3)\}$}. The statement {\(\Sigma_6\)} proves the following implication: if the equation {\(x!+1=y^2\)} has only finitely many solutions in positive integers \(x\) and \(y\), then each such solution {$(x,y)$} belongs to the set {$\{(4,5),(5,11),(7,71)\}$}. The statement {\(\Sigma_9\)} implies the infinitude of primes of the form {\(n^2+1\)}. The statement {\(\Sigma_9\)} implies that any prime of the form {\(n!+1\)} with {\(n \geqslant 2^{\textstyle 2^{9-3}}\)} proves the infinitude of primes of the form {\(n!+1\)}. The statement {\(\Sigma_{14}\)} implies the infinitude of twin primes. The statement {\(\Sigma_{16}\)} implies the infinitude of Sophie Germain primes.