MathDB

There exist

4

positive integers

a,b,c,d

such that

abcd \neq 1

and each pair of them have a GCD of

1

. Two functions

f,g : \mathbb{N} \rightarrow \{0,1\}

are multiplicative functions such that for each positive integer

n

we have :

f(an+b)=g(cn+d)

Prove that at least one of the followings hold.

i)

for each positive integer

n

we have

f(an+b)=g(cn+d)=0

ii)

There exists a positive integer

k

such that for all

n

where

(n,k)=1

we have

g(n)=f(n)=1

(Function

f

is multiplicative if for any natural numbers

a,b

we have

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

)
Proposed by Navid Safaii

Problem Statement