Partition regularity of generalised Fermat equations

TL;DR

This paper proves that in any finite coloring of a prime field, there are many solutions to the generalized Fermat equation with all variables of the same color, extending known results to new cases.

Contribution

It establishes the existence of numerous monochromatic solutions to generalized Fermat equations in prime fields, including new cases such as x + y = z^2.

Findings

More than c_{r,α,β,γ} p^2 solutions in any r-coloring of _p

Results apply to equations like x + y = z^2, previously unproven in this context

Provides quantitative bounds depending on the number of colors and exponents

Abstract

Let $\alpha,\beta,\gamma\in\mathbb{N}$. We prove that given an $r$-colouring of $\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\alpha,\beta,\gamma} p^2$ solutions to the equation $x^\alpha+y^\beta=z^\gamma$ with all of $x,y,z$ of the same colour. Here $c_{r,\alpha,\beta,\gamma}>0$ is some constant depending on the number of colours and the exponents in the equation. This is already a new result for $\alpha=\beta=1$ and $\gamma=2$, that is to say for the equation $x+y=z^2$.