Paradoxical behavior in Collatz sequences

TL;DR

This paper investigates paradoxical behaviors in Collatz sequences, linking them to the conjecture and suggesting such behaviors are rare, thus supporting Terras' hypothesis about sequence properties.

Contribution

It demonstrates that paradoxical Collatz sequences are likely finite and closely related to the conjecture, providing new insights into sequence behavior and supporting Terras' conjecture.

Findings

Paradoxical sequences are likely finite in number.

Such behaviors are closely related to the Collatz conjecture.

Results support Terras' conjecture about sequence properties.

Abstract

On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial cycle $(1,2)$. In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting with an integer $n\geq2$ is sufficient to determine its stopping time, namely, the number of iterations needed to descend below $n$. However, when iterating beyond the stopping time, there exist "paradoxical" sequences of finite length whose first term is unexpectedly exceeded, given the proportion of odd terms. In the present study, we show that this non-typical behavior is closely related to the Collatz conjecture. Furthermore, we find that it most likely occurs finitely many times, thus lending support to Terras' conjecture.