On sets W \subseteq N\{0} for which we can compute t(W) \in N such that any element of W which is greater than t(W) proves that W is infinite

Abstract : Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n \geq 2. For a positive integer n, let \Gamma_n denote the statement: if a system S \subseteq {x_i!=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in \{1,...,n}} has at most finitely many solutions in integers x_1,...,x_n greater than 1, then each such solution (x_1,...,x_n) satisfies min(x_1,...,x_n) \leq f(n). We conjecture that the statements \Gamma_1,...,\Gamma_{16} are true. The statement \Gamma_9 proves the implication: if there exists an integer x>f(9) such that x^2+1 is prime, then there are infinitely many primes of the form n^2+1. The statement Gamma_{16} proves the implication: if there exists a twin prime greater than f(16)+3, then there are infinitely many twin primes. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for every positive integer n. We formulate a conjecture which proves the implication: if 2^{2^n}+1 is composite for some integer n>g(13), then 2^{2^n}+1 is composite for infinitely many positive integers n.
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Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01614087
Contributeur : Apoloniusz Tyszka <>
Soumis le : lundi 18 décembre 2017 - 03:47:13
Dernière modification le : samedi 6 janvier 2018 - 01:01:48

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Apoloniusz Tyszka. On sets W \subseteq N\{0} for which we can compute t(W) \in N such that any element of W which is greater than t(W) proves that W is infinite. 2017. 〈hal-01614087v3〉

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