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# A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

Version 1
: Received: 10 April 2018 / Approved: 10 April 2018 / Online: 10 April 2018 (09:18:13 CEST)

Version 2 : Received: 13 April 2018 / Approved: 16 April 2018 / Online: 16 April 2018 (05:12:59 CEST)

Version 2 : Received: 13 April 2018 / Approved: 16 April 2018 / Online: 16 April 2018 (05:12:59 CEST)

How to cite:
Tyszka, A. A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions. *Preprints* **2018**, 2018040121 (doi: 10.20944/preprints201804.0121.v1).
Tyszka, A. A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions. Preprints 2018, 2018040121 (doi: 10.20944/preprints201804.0121.v1).

## Abstract

Let $f\left(1\right)=1$ , and let $f(n+1)={2}^{{\textstyle {2}^{{\textstyle f\left(n\right)}}}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {

*x*_{i }·*x*_{j }=*x*_{k }:*i*,*j*,*k*∈ {1, . . . ,*n*}} ∪ {*x**+ 1 =*_{i}*x*_{k }:*i*,*k*∈{1, . . . ,*n*}} has only finitely many solutions in non-negative integers*x*_{1}, . . . ,*x*_{n}, then each such solution (*x*_{1}, . . . ,*x*_{n}) satisfies*x*_{1}, . . . ,*x*_{n }≤*f*(2*n*). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ $\mathbb{N$ $}$ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.## Subject Areas

computable upper bound on the heights of rational solutions; computable upper bound on the moduli of integer solutions; Diophantine equation with a finite number of solutions; finite-fold Diophantine representation; single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; single query to an oracle that decides whether or not a given Diophantine equation has a rational solution

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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