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

Published:2018-01-17

• Ben Joseph Green,
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, UK
• Sofia Lindqvist,
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, UK
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## 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.