Small systems of Diophantine equations which have only very large integer solutions

TL;DR

This paper constructs specific small systems of Diophantine equations with only very large integer solutions, demonstrating the existence of systems with infinitely many solutions confined to extremely large bounds.

Contribution

It introduces an algorithm that, for any computable function, produces systems with solutions constrained to very large integers, and explicitly constructs such systems for n>=12.

Findings

Systems have infinitely many solutions within huge bounds

Solutions are confined to exponentially large ranges

Constructs applicable for all n >= 12

Abstract

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a system S \subseteq E_n such that S has infinitely many integer solutions and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n). For each integer n>=12 we construct a system S \subseteq E_n such that S has infinitely many integer solutions and they all belong to Z^n\[-2^{2^{n-1}},2^{2^{n-1}}]^n.