Question 36
Prove that it is possible to colour each positive integer with one of three colours so that the following conditions are satisfied:
i) For each $n \in \mathbb{N}_0$, all positive integers $x$ such that $2^n \le x \lt 2^{n+1}$ have the same colour.
ii) There are no positive integers $x$, $y$ and $z$ of the same colour (except $x = y = z = 2$) such that $x+y=z^2$. (B. Green, S. Lindqvist, arXiv:1608.08374)
i) For each $n \in \mathbb{N}_0$, all positive integers $x$ such that $2^n \le x \lt 2^{n+1}$ have the same colour.
ii) There are no positive integers $x$, $y$ and $z$ of the same colour (except $x = y = z = 2$) such that $x+y=z^2$. (B. Green, S. Lindqvist, arXiv:1608.08374)