On the smallest open Diophantine equations

TL;DR

This paper explores the current landscape of the simplest open polynomial Diophantine equations by ordering them by size and identifying the smallest unsolved cases across various categories, highlighting their simplicity yet difficulty.

Contribution

It introduces a systematic approach to classify and prioritize open Diophantine equations based on their size, providing a comprehensive list of the smallest unresolved equations.

Findings

Identified the smallest open equations in various categories.

Highlighted the contrast between simplicity of equations and their solution difficulty.

Provided a structured ordering of polynomial Diophantine equations.

Abstract

This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are currently open, both unrestricted and in various families, like the smallest open symmetric, 2-variable or 3-monomial equations. All the equations we discuss are amazingly simple to write down but some of them seem to be very difficult to solve.