For which sets X \subseteq \mathbb{N} the infinity of X is equivalent to the existence in X of an element that exceeds a threshold integer computed for X?

Abstract : We define computable functions g,h:N\{0} --> N\{0}. For an integer n \geq 3, let \Psi_n denote the following statement: if a system S \subseteq {x_i!=x_k: (i,k \in {1,...,n}) \wedge (i \neq k)} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq g(n). For a positive integer n, let \Gamma_n denote the following statement: if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {2^{2^{x_i}}=x_k: i,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq h(n). We prove: (1) if the equation x!+1=y^2 has only finitely many solutions in positive integers, then the statement \Psi_6 guarantees that each such solution (x,y) belongs to the set {(4,5), (5,11),(7,71)}, (2) the statement \Psi_9 proves the following implication: if there exists a positive integer x such that x^2+1 is prime and x^2+1>g(7), then there are infinitely many primes of the form n^2+1, (3) the statement \Psi_9 proves the following implication: if there exists an integer x \geq g(6) such that x!+1 is prime, then there are infinitely many primes of the form n!+1, (4) the statement \Psi_{16} proves the following implication: if there exists a twin prime greater than g(14), then there are infinitely many twin primes, (5) the statement \Gamma_{13} proves the following implication: if n \in N\{0} and 2^{2^n}+1 is composite and greater than h(12), then 2^{2^n}+1 is composite for infinitely many positive integers n.
Type de document :
Pré-publication, Document de travail
2018
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https://hal.archives-ouvertes.fr/hal-01614087
Contributeur : Apoloniusz Tyszka <>
Soumis le : mardi 30 octobre 2018 - 02:15:28
Dernière modification le : vendredi 9 novembre 2018 - 01:02:32

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  • HAL Id : hal-01614087, version 6

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Apoloniusz Tyszka. For which sets X \subseteq \mathbb{N} the infinity of X is equivalent to the existence in X of an element that exceeds a threshold integer computed for X?. 2018. 〈hal-01614087v6〉

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