MathDB

Find the smallest positive integer

k

for which there exists a colouring of the positive integers

\mathbb{Z}_{>0}

with

k

colours and a function

f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}

with the following two properties:

(i)

For all positive integers

m,n

of the same colour,

f(m+n)=f(m)+f(n).

(ii)

There are positive integers

m,n

such that

f(m+n)\ne f(m)+f(n).

In a colouring of

\mathbb{Z}_{>0}

with

k

colours, every integer is coloured in exactly one of the

k

colours. In both

(i)

and

(ii)

the positive integers

m,n

are not necessarily distinct.

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