Clusters of Integers with Equal Total Stopping Times in the 3x + 1 Problem

TL;DR

This paper introduces a recursive function to analyze the clustering of integers with equal total stopping times in the 3x + 1 problem, providing new conditions and results that deepen understanding of integer trajectories.

Contribution

It presents a simple recursive function C(n) that characterizes when consecutive integers have coinciding trajectories, offering new and existing results on total stopping times.

Findings

Derived necessary and sufficient conditions for trajectory coincidence

Reproduced known total stopping time equalities from literature

Established several novel results on integer clustering in the 3x + 1 problem

Abstract

The clustering of integers with equal total stopping times has long been observed in the 3x + 1 Problem, and a number of elementary results about it have been used repeatedly in the literature. In this paper we introduce a simple recursively defined function C(n), and we use it to give a necessary and sufficient condition for pairs of consecutive even and odd integers to have trajectories which coincide after a specific pair-dependent number of steps. Then we derive a number of standard total stopping time equalities, including the ones in Garner (1985), as well as several novel results.