Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 28, 2022

Copyright © The Korean Mathematical Society.

Multiplicative functions commutable with binary quadratic forms \(x^2 \pm xy + y^2\)

Poo-Sung Park

Kyungnam University

Abstract

If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e.,
\[
f(x^2+xy+y^2) = f(x)^2 + f(x)\,f(y) + f(y)^2,
\]
then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form $x^2-xy+y^2$, then $f$ is one of three kinds of functions: the identity function, the constant function, and an indicator function for $\mathbb{N}\setminus p\mathbb{N}$ with a prime $p$.

Keywords: additive uniqueness, multiplicative function, functional equation, quadratic form

MSC numbers: 11A25, 11E20

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