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 $\alpha ,\beta \in {\mathbb{Z}}_{>0}$ so that $\alpha$ and $\beta$ are coprime. Let $\overline{k}=\left(k_1,\dots ,k_{\beta -1}\right)$ so that for each $1\le i\le \beta -1$ , $k_i\in {\mathbb{Z}}_{>0}$ , $k_i$ is coprime to $\alpha$ and $\beta$ , and $k_i\equiv i\left(\mathrm{mod}\beta \right)$ . We define the function $C_{\left(\alpha ,\beta ,\overline{k}\right)}:{\mathbb{Z}}_{>0}⟶{\mathbb{Z}}_{>0}$ and the sequence $\left\{n,C_{\left(\alpha ,\beta ,\overline{k}\right)}\left(n\right),C_{\left(\alpha ,\beta ,\overline{k}\right)}^{}\right\}$