On ZFC-formulae \phi(x) for which we know a non-negative integer n such that max({x \in N: \phi(x)}) \leq n if the set {x \in N: \phi(x)} is finite


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Abstract
Let \Gamma(k) denote (k-1)!, and let \Gamma_{n}(k) 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.
Keywords Brocard's problem  Brocard-Ramanujan equation x!+1=y^2  composite Fermat numbers  Erdos' equation x(x+1)=y!  prime numbers of the form n^2+1  prime numbers of the form n!+1  Sophie Germain primes  twin primes
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Definability and Decision Problems in Arithmetic.Julia Robinson - 1949 - Journal of Symbolic Logic 14 (2):98-114.

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