A hypothetical way to compute an upper bound for the heights of solutions of a Diophantine equation with a finite number of solutions

TL;DR

The paper proposes a conjecture that provides an explicit upper bound for solutions of certain Diophantine equations with finitely many solutions, and discusses the implications for algorithmically bounding solution heights.

Contribution

It introduces a specific function conjectured to bound solutions of particular Diophantine systems and shows this leads to an algorithm for estimating solution heights if the conjecture holds.

Findings

The function f cannot be decreased.

The conjecture implies an algorithm exists for bounding solution heights.

Partial confirmation of the conjecture is possible via brute-force algorithms.

Abstract

Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is 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. We show that if the conjecture is true, then this can be partially confirmed by the execution of a brute-force algorithm.