On sets X \subseteq N whose finiteness implies that we know an algorithm which for every n \in N decides the inequality max(X)

Apoloniusz Tyszka (1)

1 University of Agriculture in Krakow

Let Γ_{n}(k) denote (k−1)!, where n∈{3,…,16} and k \in {2} \cup [2^{2^{n-3}}+1,\infty) \cap N. 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}} 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 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.

Mots clés: Brocard-Ramanujan equation x!+1=y^2; Erdos' equation x(x+1)=y!; twin prime conjecture; prime numbers of the form n^2+1; composite Fermat numbers; prime numbers of the form n!+1; Sophie Germain primes; Brocard's problem

Domaines: Théorie des nombres [math.NT]; Logique [math.LO]

Apoloniusz Tyszka. On sets X \subseteq N whose finiteness implies that we know an algorithm which for every n \in N decides the inequality max(X)⟨hal-01614087v7⟩