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)
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)
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 ${\mathcal X} \subseteq$ $\mathbb N$ Whose Finiteness Implies That We Know an Algorithm Which for Every $n \in \mathbb N$ Decides the Inequality $\max(\mathcal X)Preprints 2018, 2018110301
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 $\max(\mathcal X). Preprints 2018, 2018110301
Abstract
Let {$\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.
Keywords
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
Subject
Computer Science and Mathematics, Logic
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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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)
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 ${\mathcal X} \subseteq$ $\mathbb N$ Whose Finiteness Implies That We Know an Algorithm Which for Every $n \in \mathbb N$ Decides the Inequality $\max(\mathcal X)
Abstract
Keywords
Subject
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
