Multiplicative functions additive on polygonal numbers

We prove that the set $$\mathcal {P}$$ ( $$\mathcal {H}$$ , resp.) of all positive pentagonal (hexagonal, resp.) numbers is an additive uniqueness set for the collection of multiplicative functions; if a multiplicative function f satisfies the equation $$\begin{aligned} f(a+b) = f(a) + f(b) \end{aligned}$$ for all $$a, b \in \mathcal {P}$$ ( $$\mathcal {H}$$ , resp.), then f is the identity function.