A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition

In this paper, we study the bounded trajectories of Collatz-like functions. Fix α,β∈Z>0 so that α and β are coprime. Let k¯=k1,…,kβ−1 so that for each 1≤i≤β−1, ki∈Z>0, ki is coprime to α and β, and ki≡i mod β. We define the function Cα,β,k¯:Z>0⟶Z>0 and the sequence n,Cα,β,k¯n,Cα,β,k¯2n,⋯ a trajectory of n. We say that the trajectory of n is an integral loop if there exists some N in Z>0 so that Cα,β,k¯Nn=n. We define the characteristic mapping χα,β,k¯:Z>0⟶0,1,…,β−1 and the sequence n,χα,β,k¯n,χα,β,k¯2n,⋯ the characteristic trajectory of n. Let B∈Zβ be a β-adic sequence so that B=χα,β,k¯ini≥0. We say that B is eventually periodic if it eventually has a purely β-adic expansion. We show that the trajectory of n eventually enters an integral loop if and only if B is eventually periodic.