Partition regularity of generalised Fermat equations

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 $x+y=z^2$.

arXiv comment24 pages; suggestions from referees added in v3