MathDB

Let

\mathbb{N}=\{1, 2, 3, \dots\}

be the set of all positive integers. Find all functions

f : \mathbb{N} \rightarrow \mathbb{N}

such that for any positive integers

a

and

b

, the following two conditions hold: (1)

f(ab) = f(a)f(b)

, and (2) at least two of the numbers

f(a)

f(b)

, and

f(a+b)

are equal.

2