# On sets X \subseteq \mathbb{N} for which we know an algorithm that computes a threshold number t(X) \in \mathbb{N} such that X is infinite if and only if X contains an element greater than t(X)

Abstract
Let \Gamma_{n} denote (k-1)!, where n \in {3,...,16} and k \in {2} \cup [2^{2^{n-3}}+1,\infty) \cap N. For an integer n \in {3,...,16}, let \Sigma_n denote the following statement: if a system of equations S \subseteq {\Gamma_{n}(x_i)=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} with Gamma instead of \Gamma_n has only finitely many solutions in positive integers x_1,...,x_n, then every tuple (x_1,...,x_n) \in (N\{0})^n that solves the original system S satisfies x_1,...,x_n \leq 2^{2^{n-2}}. Our hypothesis claims that the statements \Sigma_{3},...,\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 \geq 2^{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. A modified statement \Sigma_7 implies the infinitude of Wilson primes.
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First archival date: 2017-10-17
Latest version: 55 (2019-01-30)
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