The colour of numbers

Consider all natural numbers, i.e. $$\{0,1,2,\dots\}$$, and suppose you are to colour each of these numbers using no more than three different colours.
Is it possible to colour all numbers, such that for each two different numbers, their sum and product have different colours?

It's not possible when using only three colours.

Suppose it is possible; let's say we're using the colours red, blue and green.
As for each non-zero natural number $$n$$, $$0\cdot n=0$$ and $$0+n=n$$, $$0$$ has to get a colour different from any other natural number. Without loss of generality, let $$0$$ be green. As a consequence, all other numbers must be either red or blue.
Furthermore, for each non-zero natural number $$m$$, $$1\cdot m=m$$ must have a different colour than $$1+m$$. This leaves us to colour all non-zero odd numbers red and all non-zero even numbers blue.
We get our contradiction by taking two different, non-zero, even numbers, e.g. $$2$$ and $$4$$. Their sum is $$6$$, while their product is $$8$$. These two numbers are even, and thus have the same colour.

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