# 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 Γ_{n}(k) denote (k−1)!, where n∈{3,…,16} and k∈{2}∪{2^{2^{n−3}}+1, 2^{2^{n−3}}+2, 2^{2^{n−3}}+3,…}. For an integer n∈{3,…,16}, let Σ_n denote the following statement: if a system of equations S⊆{Γ_{n}(x_i)=x_k: i,k∈{1,…,n}}∪{x_i⋅x_j=x_k: i,j,k∈{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⩽2^{2^{n−2}}. The statement Σ_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 Σ_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 Σ_9 implies the infinitude of primes of the form n^2+1. The statement Σ_9 implies that any prime of the form n!+1 with n⩾2^{2^{9−3}} proves the infinitude of primes of the form n!+1. The statement Σ_{14} implies the infinitude of twin primes. The statement Σ_{16} implies the infinitude of Sophie Germain primes. We formulate a hypothesis which implies the infinitude of Wilson primes.
Keywords :
Type de document :
Pré-publication, Document de travail
2019
Domaine :

https://hal.archives-ouvertes.fr/hal-01614087
Contributeur : Apoloniusz Tyszka <>
Soumis le : mardi 8 janvier 2019 - 06:59:21
Dernière modification le : samedi 19 janvier 2019 - 01:02:51

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• HAL Id : hal-01614087, version 7

### Citation

Apoloniusz Tyszka. 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). 2019. 〈hal-01614087v7〉

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