MathDB

Prove that it is possible to color each positive integers with one of three colors 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<2^{n+1}

have the same color.

ii)

There are no positive integers

x,y,z

of the same color (except

x=y=z=2

) such that

x+y=z^2.

Problem Statement