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Multiplicative functions additive on polygonal numbers
Aequationes Mathematicae ( IF 0.9 ) Pub Date : 2021-06-28 , DOI: 10.1007/s00010-021-00824-8 Byungchan Kim , Ji Young Kim , Chong Gyu Lee , Poo-Sung Park
中文翻译:
多边形数上的乘法函数
更新日期:2021-06-28
Aequationes Mathematicae ( IF 0.9 ) Pub Date : 2021-06-28 , DOI: 10.1007/s00010-021-00824-8 Byungchan Kim , Ji Young Kim , Chong Gyu Lee , Poo-Sung Park
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:
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。