Problem

Source: Iran 2024 3rd Round Test 1 P1

Tags: algebra, complex numbers, fe, function, functional equation



Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$Where $\overline{a}$ is the complex conjugate of $a$.