On the equation \(G(n)=F(n^2-1)+D\)

Abstract. We give all solutions \((D, G, F)\) of the equation $$G(n)=F(n^2-1)+D\quad\text{for every}\quad n\in \mathbb{N},$$ where \(G, F\) are completely multiplicative functions and \(D\in\mathbb{C}\).