On the Link between Stopping Time and Non-Trivial Cycles in the Collatz Problem

Abstract

The Collatz Conjecture asserts that for all positive integers s , every Syracuse integer sequence defined by T ( s ) = s / 2 if s is even, and T ( s ) = ( 3 s + 1 ) / 2 otherwise, eventually reaches 1 after a finite number of iterations. The stopping time of an integer is the smallest number of iterations required for the sequence to fall below its starting value, while the total stopping time measures the iterations needed to reach 1. In this paper, we revisit the notion of stopping time by introducing the coefficient stopping time, defined as the smallest value of n such that the coefficient of s in T n ( s ) , expressed as 3 r / 2 n , is less than 1. Building on foundational results by Lynn E. Garner (1981), we leverage recent computational results by David Barina to extend Garner’s estimation regarding the minimal length of non-trivial cycles. Specifically, we demonstrate the non-existence of non-trivial cycles of length n < 19478780533 , thus improving upon the previous result by Shalom Eliahou (2021). We subsequently show that this result can be generalized to all integers n . We also introduce new properties concerning the behavior of Syracuse sequences modulo 2 n , which play a central role in our approach. Inspired by the work of Mike Winkler (2017), we provide an exact formulation of the stopping time counting function, which calculates the number of integers s < 2 n whose stopping time σ ( s ) = n . From this formulation, we demonstrate that the density of integers with stopping time greater than n tends to zero as n approaches infinity. Furthermore, if divergent sequences exist, the set of such sequences is of zero density in ℕ . Our results offer a deeper understanding of how stopping time behavior relates to the elusive search for non-trivial cycles in the Collatz problem.