Diophantine equations: a systematic approach

TL;DR

This paper introduces a systematic approach to classifying and solving polynomial Diophantine equations based on a new size measure, successfully solving many cases and identifying challenging open problems to inspire future research.

Contribution

It defines a novel size measure for Diophantine equations, orders them accordingly, and solves numerous cases using a combined computer-aided and human reasoning approach.

Findings

Solved Hilbert's tenth problem for equations of size less than 31

Identified the smallest open equations in various categories

Compiled a list of simple yet difficult-to-solve equations

Abstract

This paper initiates a novel research direction in the theory of Diophantine equations: define an appropriate version of the equation's size, order all polynomial Diophantine equations starting from the smallest ones, and then solve the equations in that order. By combining a new computer-aided procedure with human reasoning, we solved the Hilbert's tenth problem for all polynomial Diophantine equations of size less than $31$, where the size is defined in (Zidane, 2018). In addition, we solved this problem for all equations of size equal to $31$, with a single exception. Further, we solved the Hilbert's tenth problem for all two-variable Diophantine equations of size less than $32$, all symmetric equations of size less than $39$, all three-monomial equations of size less than $45$, and, in each category, identified the explicit smallest equations for which the problem remains open. As a result, we derived a list of equations that are very simple to write down but which are apparently difficult to solve. As we know from the example of Fermat's Last Theorem, such equations have a potential to stimulate the development of new methods in number theory.