Multiplicative functions preserving the sum-of-three-squares operation

TL;DR

This paper proves that any multiplicative function preserving the sum of three squares must be the identity function, highlighting a unique structural property of such functions.

Contribution

It establishes a rigidity result for multiplicative functions that preserve the sum-of-three-squares operation, showing they are necessarily the identity.

Findings

Any multiplicative function satisfying the sum-of-three-squares preservation is the identity function.

The result characterizes the structure of functions preserving a specific quadratic form.

The proof confirms the uniqueness of the identity function under these conditions.

Abstract

If a multiplicative function $f$ satisfies $f(a^2+b^2+c^2) = f(a)^2+f(b)^2+f(c)^2$ for all positive integers $a$, $b$, and $c$, then $f$ is an identity function.