  Green and Lindqvist proved that for any 2-colouring of $\mathbb{N}$, there
are in\-fi\-ni\-tely many monochromatic solutions to $x+y=z^2$. In fact, they
showed the existence of a monochromatic solution in every interval $[N,cN^8]$
with large enough $N$. In this short note we give a different proof for their
theorem and prove that a monochromatic solution exists in every interval
$[N,10^4N^4]$ with large enough $N$. A 2-colouring of $[N,(1/27)N^4]$ avoiding
monochromatic solutions to $x+y=z^2$ is also presented, which shows that in
$10^4N^4$ only the constant factor can be reduced.
