Preprint Article Version 5 This version is not peer-reviewed

On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$

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 ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$. Preprints 2018, 2018110301 (doi: 10.20944/preprints201811.0301.v5). Tyszka, A. On Sets ${\mathcal X} \subseteq$ $\mathbb N$ for Which We Know an Algorithm That Computes a Threshold Number $t({\mathcal X}) \in$ $\mathbb N$ Such That ${\mathcal X}$ Is Infinite If and Only If ${\mathcal X}$ Contains an eLement Greater Than $t({\mathcal X})$. Preprints 2018, 2018110301 (doi: 10.20944/preprints201811.0301.v5).

Abstract

Let $\Gamma_{\fbox{n}}(k)$ denote $(k-1)!$, where $n \in \{3,\ldots,14\}$ and $k \in \{2\} \cup \left\{2^{\textstyle 2^{n-3}}+1, 2^{\textstyle 2^{n-3}}+2, 2^{\textstyle 2^{n-3}}+3, \ldots\right\}$. For an integer $n \in \{3,\ldots,14\}$, let $\Sigma_n$ denote the following statement: if a system of equations ${\mathcal S} \subseteq \{\Gamma_{\fbox{n}}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}$. 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.

Subject Areas

Brocard's problem; Brocard-Ramanujan equation $x!+1=y^2$; composite Fermat numbers; Erdös' equation $x(x+1)=y!$; prime numbers of the form $n^2+1$; prime numbers of the form $n!+1$; Richert's lemma; twin prime conjecture