A systematic approach to Diophantine equations: open problems
Bogdan Grechuk
TL;DR
This work aggregates the current landscape of small, simple-to-write polynomial Diophantine equations that remain open, introducing a size measure $H(P)=\sum|a_i|2^{d_i}$ and a framework for classifying equations by symmetry and monomial structure. It formalizes key questions about parametrizations, descriptions of all integer or rational solutions, and the finiteness of the solution set, cataloguing the smallest open cases across unrestricted, homogeneous, symmetric, cyclic, and multi-monomial categories. The document also formalizes the problem of arbitrarily large solutions and the existence of integer solutions (Hilbert’s 10th problem) with explicit open instances, and discusses the notion of the shortest open equations via a length metric $l(P)$. Finally, it tracks the evolution of these open problems across versions, noting which equations have been solved and how that changes the open problem landscape. Overall, the paper serves as a structured reference for researchers tracking open questions in polynomial Diophantine equations and their classifications.
Abstract
This paper collects polynomial Diophantine equations that are amazingly simple to write down but are apparently difficult to solve.