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n = (f(2n)-f(n) )(2 f(n) - f(2n) )

Source: 2021 Francophone MO Seniors p4

April 3, 2021
number theoryfunctional equationfunctionalalgebraFrancophone

Problem Statement

Let N≥1\mathbb{N}_{\ge 1} be the set of positive integers. Find all functions f ⁣:N≥1→N≥1f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1} such that, for all positive integers mm and nn:
(a) n=(f(2n)−f(n))(2f(n)−f(2n))n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right), (b)f(m)f(n)−f(mn)=(f(2m)−f(m))(2f(n)−f(2n))+(f(2n)−f(n))(2f(m)−f(2m))f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right), (c) m−nm-n divides f(2m)−f(2n)f(2m)-f(2n) if mm and nn are distinct odd prime numbers.
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