Monochromatic solutions to $x + y = z^2$

TL;DR

The paper investigates monochromatic solutions to the equation x + y = z^2 under different colorings of natural numbers, proving infinite solutions exist with two colors but not with three.

Contribution

It establishes the existence of infinitely many monochromatic solutions in 2-colorings and the finiteness in some 3-colorings, advancing understanding of coloring solutions to quadratic equations.

Findings

Infinite monochromatic solutions in 2-colorings

Existence of 3-colorings with finitely many solutions

Differentiates solution behavior between 2 and 3 colors

Abstract

Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely many monochromatic solutions to $x + y = z^2$. On the other hand, there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation.