Additive Collatz Trajectories

TL;DR

This paper investigates Additive Collatz Trajectories, providing conditions for looping, and introduces an algorithm to classify natural numbers based on their trajectory behavior.

Contribution

It offers a necessary and sufficient condition for trajectory looping and proposes an algorithm to analyze equivalence classes of natural numbers.

Findings

Established a condition for looping in Additive Collatz Trajectories

Developed an algorithm to compute equivalence classes

Presented two minor results related to trajectory behavior

Abstract

Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study. This paper explores Additive Collatz Trajectories, a particular case of a generalization of Collatz conjecture and puts forward a sufficient and necessary condition for looping of Additive Collatz Trajectories, along with two minor results. An algorithm to compute the number of equivalence classes when natural numbers are quotiented by the limiting behavior of their corresponding trajectories is also proposed.