Clusters of Integers with Equal Total Stopping Times 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.