Paradoxical behavior in Collatz sequences
O Rozier, C Terracol - arXiv preprint arXiv:2502.00948, 2025 - arxiv.org
O Rozier, C Terracol
arXiv preprint arXiv:2502.00948, 2025•arxiv.orgOn the set of positive integers, we consider an iterated process that sends $ n $ to $\frac
{3n+ 1}{2} $ or to $\frac {n}{2} $ depending on the parity of $ n $. According to a conjecture
due to Collatz, all such sequences end up in the cycle $(1, 2) $. In a seminal paper, Terras
further conjectured that the proportion of odd terms encountered when starting from $ 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" …
{3n+ 1}{2} $ or to $\frac {n}{2} $ depending on the parity of $ n $. According to a conjecture
due to Collatz, all such sequences end up in the cycle $(1, 2) $. In a seminal paper, Terras
further conjectured that the proportion of odd terms encountered when starting from $ 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" …
On the set of positive integers, we consider an iterated process that sends to or to depending on the parity of . According to a conjecture due to Collatz, all such sequences end up in the cycle . In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting from is sufficient to determine its stopping time, namely, the number of iterations needed to descend below . However, when iterating beyond the stopping time, there exist "paradoxical" sequences for which the first term is unexpectedly exceeded. In the present study, we show that this topic is strongly linked to the Collatz conjecture. Moreover, this non-typical behavior seems to occur finitely many times apart from the trivial cycle, thus lending support to Terras' conjecture.
arxiv.org