Additive uniqueness sets for a pair of multiplicative functions

Abstract. We give all solutions of those multiplicative functions \(f,g\) which satisfy $$ f\left(n^2+m^2+a+b\right)=g(n^2+a)+g(m^2+b)\quad\hbox{for all}\quad n,m\in{\Bbb N}, $$ where \(a,b\) are non-negative integers with \(a+b>0\). It is proved that if $$ g(a+36)+4g(a+25)-g(a+9)-g(a+4)-3g(a+1)\neq 0, $$ then $$ f(n)=n \quad\text{and}\quad g(m^2+a)=m^2+a,~ g(m^2+b)=m^2+b $$ for all \(n, m\in{\Bbb N}, ~(n,2(a+b))=1\).