Handles, Hooks, and Scenarios: A fresh Look at the Collatz Conjecture

Abstract

An operational approach to the Collatz Conjecture is presented. Scenarios are defined as strings of characters "s" (for "spike") and "d" (for "down") which symbolize the Collatz operations (3m+1)/2 and m/2 in a Collatz Series connecting two odd integers, the startnumber and the endnumber. It is shown that a scenario determines uniquely four integers, called startperiod, startphase, endperiod, and endphase such that startnumber = startperiod x k minus startphase, and endnumber = endperiod x k minus endphase, where k is any natural number. Therefore, any scenario can be realized infinitely many times, with well-defined startnumber and endnumber for each realization. It is shown how the periods and phases for a given scenario are calculated. The results are used to prove that any odd (even) number in a Collatz Series is less than 8 (7) ("up"- or "down"-) steps away from an odd integer which is divisible by 3. They are also used for the construction of Collatz Series which exhibit prescribed regular graphics patterns. Finally, the bearing of the present work on the question of non-trivial cycles is discussed.