MathDB

Number theory:function

Source: Iran 3rd round 2017 Number theory first exam-P3

August 9, 2017

number theoryfunction

Problem Statement

Let

k

be a positive integer. Find all functions

f:\mathbb{N}\to \mathbb{N}

satisfying the following two conditions:\\
• For infinitely many prime numbers

p

there exists a positve integer

c

such that

f(c)=p^k

.\\
• For all positive integers

m

and

n

f(m)+f(n)

divides

f(m+n)