A characterization of functions using Lagrange's
Four-Square Theorem

Abstract. We give all solutions of the arithmetical functions \(f, g, F, G: {\mathbb N}_0\to\mathbb{C}\) which satisfy the relations $$ f\bigl(a^2+b^2+c^2+d^2\bigr)=g\bigl(a^2\bigr)+g\bigl(b^2\bigr)+ g\bigl(c^2\bigr)+g\bigl(d^2\bigr) $$ and $$ F\bigl(a^2+b^2+c^2+d^2\bigr)=G\bigl(a^2+b^2\bigr)+G\bigl(c^2+d^2\bigr) $$ for every \(a,b,c,d\in{\mathbb N}_0\), where \({\mathbb N}_0\) is the set of all non-negative integers.