Bulletin of the
Korean Mathematical Society BKMS

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: Primary 11A25, 11E20

Supported by: This work was supported by Kyungnam University Foundation Grant, 2019.