Preprint Article Version 2 This version is not peer-reviewed

# 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)

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.v2). 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.v2).

## Abstract

Let $f\left(1\right)=1$ , and let $f\left(n+1\right)={2}^{{2}^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn f (2n). 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