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.

中文翻译：

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多边形数上的乘法函数

我们证明所有正五边形（六边形，分别）数的集合\(\mathcal {P}\) ( \(\mathcal {H}\) , resp.) 是乘法函数集合的加性唯一性集; 如果乘法函数f满足方程

$$\begin{aligned} f(a+b) = f(a) + f(b) \end{aligned}$$

对于所有\(a, b \in \mathcal {P}\) ( \(\mathcal {H}\) , resp.)，则f是恒等函数。