Fermat-like equations that are not partition regular

TL;DR

This paper identifies a broad class of Fermat-like Diophantine equations that are not partition regular, using elementary coefficient conditions, including examples like $x^n+y^m=z^k$ with $k$ outside the set of exponents.

Contribution

It introduces elementary coefficient conditions to determine non-partition regularity of Fermat-like equations, expanding understanding of their combinatorial properties.

Findings

Large class of Fermat-like equations are not partition regular

Elementary conditions effectively identify non-partition regular equations

Includes examples like $x^n+y^m=z^k$ with $k\notin\{n,m\}$.

Abstract

By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being $x^n+y^m=z^k$ with $k\notin\{n,m\}$.