A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

TL;DR

This paper explores a conjecture that proposes a bound on the solutions of certain Diophantine equations, leading to implications for the decidability of finiteness and computability of solution sets.

Contribution

It introduces a conjecture on solution bounds for specific Diophantine systems and shows its implications for algorithmic decidability and computability in number theory.

Findings

The conjecture implies an algorithm for bounding solutions of finite Diophantine equations.

It suggests decidability of finiteness for rational solutions with an oracle.

It indicates that sets with finite-fold Diophantine representations are computable.

Abstract

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {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 \leq f(2n). We prove: (1) the conjecture implies 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 conjecture implies that the question whether or not a Diophantine equation has only finitely many rational solutions is decidable with an oracle for deciding whether or not a Diophantine equation has a rational solution, (3) the conjecture implies that the question whether or not a Diophantine equation has only finitely many integer solutions is decidable with an oracle for deciding whether or not a Diophantine equation has an integer solution, (4) the conjecture implies that if a set M \subseteq N has a finite-fold Diophantine representation, then M is computable.