Working Paper Article Version 7 This version is not peer-reviewed

# On Sets ${\mathcal X} \subseteq$ $\mathbb N$ Whose Finiteness Implies That We Know an Algorithm Which for Every $n \in \mathbb N$ Decides the Inequality $\max(\mathcal X)<n$

Version 1 : Received: 11 November 2018 / Approved: 13 November 2018 / Online: 13 November 2018 (06:58:44 CET)
Version 2 : Received: 15 November 2018 / Approved: 16 November 2018 / Online: 16 November 2018 (11:40:40 CET)
Version 3 : Received: 20 November 2018 / Approved: 20 November 2018 / Online: 20 November 2018 (06:57:27 CET)
Version 4 : Received: 29 November 2018 / Approved: 29 November 2018 / Online: 29 November 2018 (05:28:14 CET)
Version 5 : Received: 29 December 2018 / Approved: 3 January 2019 / Online: 3 January 2019 (09:43:19 CET)
Version 6 : Received: 9 January 2019 / Approved: 10 January 2019 / Online: 10 January 2019 (07:08:38 CET)
Version 7 : Received: 5 April 2019 / Approved: 9 April 2019 / Online: 9 April 2019 (05:40:44 CEST)
Version 8 : Received: 1 June 2019 / Approved: 5 June 2019 / Online: 5 June 2019 (08:37:55 CEST)

How to cite: Tyszka, A. On Sets ${\mathcal X} \subseteq$ $\mathbb N$ Whose Finiteness Implies That We Know an Algorithm Which for Every $n \in \mathbb N$ Decides the Inequality $\max(\mathcal X)<n$. Preprints 2018, 2018110301 Tyszka, A. On Sets&nbsp;${\mathcal X} \subseteq$ $\mathbb N$ Whose Finiteness Implies That We Know an Algorithm Which for Every&nbsp;$n \in \mathbb N$ Decides the Inequality $\max(\mathcal X)&lt;n$. Preprints 2018, 2018110301

## Abstract

Let {$\Gamma(k)$} denote {$(k-1)!$}, and let {$\Gamma_{n}(k)$} denote {$(k-1)!$}, where {$n \in \{3,\ldots,16\}$} and {$k \in \{2\} \cup [2^{\textstyle 2^{n-3}}+1,\infty) \cap \mathbb N$}. For an integer {$n \in \{3,\ldots,16\}$}, let $\Sigma_n$ denote the following statement: if a system of equations {${\mathcal S} \subseteq \{\Gamma_{n}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$} with $\Gamma$ instead of $\Gamma_{n}$ has only finitely many solutions in positive integers {$x_1,\ldots,x_n$}, then every tuple {$(x_1,\ldots,x_n) \in (\mathbb N \setminus \{0\})^n$} that solves the original system ${\mathcal S}$ satisfies {$x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}$}. Our hypothesis claims that the statements {$\Sigma_{3},\ldots,\Sigma_{16}$} 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 {$n \geqslant 2^{\textstyle 2^{9-3}}$} 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.

## Subject Areas

Brocard's problem; Brocard-Ramanujan equation $x!+1=y^2$; composite Fermat numbers; Erd\"os' 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