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

Abstract

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 ¯ = k 1 , … , k β − 1 so that for each 1 ≤ i ≤ β − 1 , k i ∈ Z > 0 , k i is coprime to α and β , and k i ≡ i mod β . We define the function C α , β , k ¯ : Z > 0 ⟶ Z > 0 and the sequence n , C α , β , k ¯ n , C α , β , k ¯ 2 n , ⋯ 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 α , β , k ¯ N n = n . We define the characteristic mapping χ α , β , k ¯ : Z > 0 ⟶ 0,1 , … , β − 1 and the sequence n , χ α , β , k ¯ n , χ α , β , k ¯ 2 n , ⋯ the characteristic trajectory of n . Let B ∈ Z β be a β -adic sequence so that B = χ α , β , k ¯ i n i ≥ 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.