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Multiplicative functions additive on polygonal numbers. (English) Zbl 1479.11023

Summary: 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 \[ f(a+b) = f(a) + f(b) \] for all \(a, b \in \mathcal{P}\) (\(\mathcal{H} \), resp.), then \(f\) is the identity function.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11E25 Sums of squares and representations by other particular quadratic forms
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