Characterization of the identity function with an equation function

Abstract. Let \(D\in\{2,3\}\) and \(f:\mathbb{N}\to\mathbb{C}\). We prove that if $$ f(n^2+Dnm+m^2)=f(n)^2+Df(n)f(m)+f(m)^2\quad \text{for every}\quad n,m\in\mathbb{N}, $$ then \(f\) is the identity function.