Clusters of Integers with Equal Total Stopping Times in the 3x + 1 Problem
Number Theory
2017-11-17 v3
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.
Keywords
Cite
@article{arxiv.1709.02979,
title = {Clusters of Integers with Equal Total Stopping Times in the 3x + 1 Problem},
author = {Mark D. LaDue},
journal= {arXiv preprint arXiv:1709.02979},
year = {2017}
}