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 is an element of Z(>0) so that alpha and beta are coprime. Let (k) over bar (k(1),..., k(beta-1)) so that for each 1 <= i <= beta - 1, k(i) is an element of Z(>0), k(i) is coprime to alpha and beta, and k(i) equivalent to i (mod beta). We define the function C (alpha,beta,(k) over bar): Z(>0) -> Z(>0) and the sequence {n, C-(alpha,C-beta,C-(k) over bar)(n), C-(alpha,beta,(k) over bar)(2)(n), center dot center dot center dot} 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 C-(alpha,beta,(k) over bar)(N) = n. We define the characteristic mapping chi((alpha,beta,(k) over bar)) : Z(>0) -> {0, 1, ..., beta-1}. and the sequence {n, chi((alpha,beta,(k) over bar))(n), chi(2)((alpha,beta,(k) over bar)) (n), center dot center dot center dot} the characteristic trajectory of n. Let B is an element of Z(beta) be a beta-adic sequence so that B. (chi(i)((alpha,beta,(k) over bar)) (n))(i >= 0). We say that B is eventually periodic if it eventually has a purely beta-adic expansion. We show that the trajectory of n eventually enters an integral loop if and only if B is eventually periodic.