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

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

A peer-reviewed article of this Preprint also exists.

Tyszka, A. A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions. Open Computer Science, Volume 8, Issue 1, Pages 109–114, Tyszka, A. A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions. Open Computer Science, Volume 8, Issue 1, Pages 109–114,

## Abstract

*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.

## Keywords

## Subject

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