Working PaperArticleVersion 9This version is not peer-reviewed

Open Problems on Computable Sets {${\mathcal X} \subseteq N$}, Which Require That a Known Integer $n$ Satisfies ${\rm card}({\mathcal X})=\omega \Leftrightarrow {\mathcal X} \cap (n,\infty) \neq \emptyset$, and Which Cannot Be Stated Formally as They Refer to Current Knowledge about ${\mathcal X}$}

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. Open Problems on Computable Sets {${\mathcal X} \subseteq N$}, Which Require That a Known Integer $n$ Satisfies ${\rm card}({\mathcal X})=\omega \Leftrightarrow {\mathcal X} \cap (n,\infty) \neq \emptyset$, and Which Cannot Be Stated Formally as They Refer to Current Knowledge about ${\mathcal X}$}. Preprints2018, 2018110301
Tyszka, A. Open Problems on Computable Sets {${\mathcal X} \subseteq N$}, Which Require That a Known Integer $n$ Satisfies ${\rm card}({\mathcal X})=\omega \Leftrightarrow {\mathcal X} \cap (n,\infty) \neq \emptyset$, and Which Cannot Be Stated Formally as They Refer to Current Knowledge about ${\mathcal X}$}. Preprints 2018, 2018110301

Cite as:

Tyszka, A. Open Problems on Computable Sets {${\mathcal X} \subseteq N$}, Which Require That a Known Integer $n$ Satisfies ${\rm card}({\mathcal X})=\omega \Leftrightarrow {\mathcal X} \cap (n,\infty) \neq \emptyset$, and Which Cannot Be Stated Formally as They Refer to Current Knowledge about ${\mathcal X}$}. Preprints2018, 2018110301
Tyszka, A. Open Problems on Computable Sets {${\mathcal X} \subseteq N$}, Which Require That a Known Integer $n$ Satisfies ${\rm card}({\mathcal X})=\omega \Leftrightarrow {\mathcal X} \cap (n,\infty) \neq \emptyset$, and Which Cannot Be Stated Formally as They Refer to Current Knowledge about ${\mathcal X}$}. Preprints 2018, 2018110301

Abstract

Let β= (((24!)!)!)!, and let {${\mathcal P}_{n^2+1}$} denote the set of all primes of the form {$n^2+1$}. Let ${\mathcal M}$ denote the set of all positive multiples of elements of the set {${\mathcal P}_{n^2+1} \cap (\beta,\infty)$}. The set {${\mathcal X}=\{0,\ldots,\beta\} \cup {\mathcal M}$} satisfies the following conditions: (1) ${\rm card}({\mathcal X})$ is greater than a huge positive integer and it is conjectured that ${\mathcal X}$ is infinite, (2)} we do not know any algorithm deciding the finiteness of ${\mathcal X}$, (3)~a~known and short algorithm for every {$n \in N$} decides whether or not {$n \in {\mathcal X}$}, (4) a known and short algorithm returns an integer~$n$ such that ${\mathcal X}$ is infinite if and only if ${\mathcal X}$ contains an element greater than $n$. The following problem is open: {\em simply define a set {${\mathcal X} \subseteq N$} such that ${\mathcal X}$ satisfies conditions (1)-(4), and \underline{we do not know any representation of~~${\mathcal X}$~~as a finite union of sets whose definitions are simpler than} \underline{the definition of ${\mathcal X}$}} {\tt (5)}. Let {$f(1)=2$}, {$f(2)=4$}, and let {$f(n+1)=f(n)!$} for every integer {$n \geqslant 2$}. For a positive integer $n$, let {$\Psi_n$} denote the following statement: {\em if a system of equations ${\mathcal S} \subseteq \Bigl\{x_i!=x_k: i,k \in \{1,\ldots,n\}\Bigr\} \cup \Bigl\{x_i \cdot x_j=x_k: i,j,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers \mbox{$x_1,\ldots,x_n$}, then each such solution \mbox{$(x_1,\ldots,x_n)$} satisfies {$x_1,\ldots,x_n \leqslant f(n)$}.} We prove that for every statement $\Psi_n$ the bound {$f(n)$} cannot be decreased. The author's guess is that the statements {$\Psi_1,\ldots,\Psi_9$} are true. We prove that the statement $\Psi_9$ implies that the set ${\mathcal X}$ of all {non-negative} integers $k$ whose number of digits belongs to {${\mathcal P}_{n^2+1}$} satisfies conditions {\tt (1)-(5)}.

Keywords

Alexander Zenkin's super-induction method; arithmetical operations on huge integers cannot be performed by any physical process; computable set {${\mathcal X} \subseteq N$} whose finiteness remains conjectured; computable set {${\mathcal X} \subseteq N$} whose infiniteness remains conjectured

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.

Commenter: Apoloniusz Tyszka

Commenter's Conflict of Interests: Author