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arXiv · 1805.06279

Monochromatic solutions to $x+y=z^2$ in the interval $[N,cN^4]$

Péter Pál Pach
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Abstract

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.

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Péter Pál Pach. 2018-05-16. Monochromatic solutions to $x+y=z^2$ in the interval $[N,cN^4]$. https://doi.org/10.1112/blms.12207

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