On conjecture concerning the functional equation

Abstract. We determine all solutions of those \(f:{\mathbb{N}}\to {\mathbb{C}}\) for which \(f(n^2+Dm^2)= f^2( n)+ Df^2(m)\) is satisfied for all positive integers \(n,m\), where \(D\) is a given positive integer. This solves a problem of Kátai and Phong.