On the equation \(F(p^k)=F(p^k-1)+1\)

Abstract. We prove that if an odd positive integer \(k\) and a completely multiplicative function \(F:\mathbb{N}\to \mathbb{C}\) satisfy the conditions \(F(p^2)=F(p^2-1)+1\) and \(F(p^k)=F(p^k-1)+1\) for every prime
\(p\), then \(F\) is the identity function. We also investigate completely functions \(F:\mathbb{N}\to\mathbb{R}\) such that \(F(p^2)=F(p^2-1)+1\) is satisfied for every prime \(p\).