MathDB

A functional equation

Source: Romanian District Olympiad 2014, Grade 9, P4

June 15, 2014

functionalgebrafunctional equationDivisibility

Problem Statement

Find all functions

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

with the properties:
[*] f(m+n) -1 \mid f(m)+f(n), \forall m,n\in\mathbb{N}^{\ast} [*]

n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast}