Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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)
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Version 5 : Received: 29 December 2018 / Approved: 3 January 2019 / Online: 3 January 2019 (09:43:19 CET)
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Version 9 : Received: 19 February 2020 / Approved: 25 February 2020 / Online: 25 February 2020 (10:06:45 CET)

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.v2). 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.v2).

## Abstract

We define computable functions $g,h:$ $\mathbb N$ $\setminus \{0\} \to$ $\mathbb N$ $\setminus \{0\}$. For an integer $n \geqslant 3$, let $\Psi_n$ denote the following statement: if a system ${\mathcal S} \subseteq \Bigl\{x_i!=x_k: (i,k \in \{1,\ldots,n\}) \wedge (i \neq k)\Bigr\} \cup \Bigl\{x_i \cdot x_j=x_k: i,j,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant g(n)$. For a positive integer $n$, let $\Gamma_n$ denote the following statement: if a system $S \subseteq \Bigl\{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\Bigr\} \cup \Bigl\{2^{\textstyle 2^{\textstyle x_i}}=x_k:~i,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers $x_1,\ldots,x_n$, then each such solution $(x_1,\ldots,x_n)$ satisfies $x_1,\ldots,x_n \leqslant h(n)$. We prove: (1) if the equation $x!+1=y^2$ has only finitely many solutions in positive integers, then the statement $\Psi_6$ guarantees that each such solution $(x,y)$ belongs to the set $\{(4,5),(5,11),(7,71)\}$, (2) the statement $\Psi_9$ proves the following implication: if there exists a positive integer $x$ such that $x^2+1$ is prime and $x^2+1>g(7)$, then there are infinitely many primes of the form $n^2+1$, (3) the statement $\Psi_9$ proves the following implication: if there exists an integer $x \geqslant g(6)$ such that $x!+1$ is prime, then there are infinitely many primes of the form $n!+1$, (4) the statement $\Psi_{16}$ proves the following implication: if there exists a twin prime greater than $g(14)$, then there are infinitely many twin primes, {\bf (5)}~the statement $\Gamma_{13}$ proves the following implication: if $n \in$ $\mathbb N$ $\setminus \{0\}$ and $2^{\textstyle 2^{\textstyle n}}+1$ is composite and greater than $h(12)$, then $2^{\textstyle 2^{\textstyle n}}+1$ is composite for infinitely many positive integers $n$.

## Subject Areas

Brocard's problem; Brocard-Ramanujan equation; composite Fermat numbers; Dickson's conjecture; halting of a Turing machine; prime numbers of the form $n^2+1$; prime numbers of the form $n!+1$; Richert's lemma; twin prime conjecture

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