All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem

TL;DR

The paper establishes that all functions with a single-fold Diophantine representation are dominated by a specific limit-computable function f, which is implemented in MuPAD, highlighting an open problem in its computability.

Contribution

It introduces a new limit-computable function f that dominates all functions with a single-fold Diophantine representation, linking Diophantine equations to computability theory.

Findings

For n ≥ 2214, systems in E_n have a unique solution exceeding 2^(2^n).

Function f(n) bounds solutions of systems with unique solutions in E_n.

f dominates any function with a single-fold Diophantine representation.

Abstract

Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any integer n \geq 2214, we define a system T \subseteq E_n which has a unique integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S \subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this solution belongs to [0,b]^n. We prove that if a function g:N-->N has a single-fold Diophantine representation, then f dominates g. We present a MuPAD code which takes as input a positive integer n, performs an infinite loop, returns a non-negative integer on each iteration, and returns f(n) on each sufficiently high iteration.