# Arithmetic functions commutable with sums of squares

## TL;DR

This paper characterizes functions from natural numbers to complex numbers that preserve the sum of squares structure for three or more variables, revealing specific functional forms that satisfy this property.

## Contribution

It provides a complete characterization of functions that commute with sums of squares for multiple variables, extending understanding of functional equations related to quadratic forms.

## Key findings

- Identifies all functions satisfying the sum of squares functional equation for k ≥ 3
- Shows the structure of such functions is highly constrained
- Provides explicit forms of functions that preserve sum of squares

## Abstract

In this note, we characterize all functions $f : \mathbb{N} \rightarrow \mathbb{C}$ such that $f(x_1^2+ \cdots + x_k^2)=f(x_1)^2+ \cdots + f(x_k)^2$, where $k \geq 3$ and $x_1, \cdots, x_k$ are positive integers.