MathDB

pmo problem 3

Source: PMO

March 19, 2021

functional equationnumber theoryPMO

Problem Statement

Denote by

\mathbb{Q}^+

the set of positive rational numbers. A function

f : \mathbb{Q}^+ \to \mathbb{Q}

satisfies
•

f(p) = 1

for all primes

p

, and
•

f(ab) = af(b) + bf(a)

for all

a,b \in \mathbb{Q}^+

.
For which positive integers

n

does the equation

nf(c) = c

have at least one solution

c

in

\mathbb{Q}^+

?

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