{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/determining-monotonic-step-equal-sequences-of","title":"Determining monotonic step-equal sequences of any limited length in the Collatz problem","arxiv_id":"1909.13218","date":"2019-09-29","proceeding":null,"authors":["Longjiang Li"],"abstract":"This paper proposes a formula expression for the well-known Collatz conjecture (or 3x+1 problem), which can pinpoint all the growth points in the orbits of the Collatz map for any natural numbers. The Collatz map $Col: \\mathcal{N}+1 \\rightarrow \\mathcal{N}+1$ on the positive integers is defined as $x_{n+1}=Col(x_n)=(3 x_n +1)/2^{m_n}$ where $x_{n+1}$ is always odd and $m_n$ is the step size required to eliminate any possible even values. The Collatz orbit for any positive integer, $x_1$, is expressed by a sequence, $<x_1$; $x_2\\doteq Col(x_1)$; $\\cdots$ $x_{n+1}\\doteq Col(x_n)$; $\\cdots>$ and $x_n$ is defined as a growth point if $Col(x_n)>x_n$ holds, and we show that every growth point is in a format of ``$4y+3$'' where $y$ is any natural number. Moreover, we derive that, for any given positive integer $n$, there always exists a natural number, $x_1$, that starts a monotonic increasing or decreasing Collatz sequence of length $n$ with the same step size. For any given positive integer $n$, a class of orbits that share the same orbit rhythm of length $n$ can also be determined.","url_abs":"https://arxiv.org/abs/1909.13218v2","url_pdf":"https://arxiv.org/pdf/1909.13218v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"determining-monotonic-step-equal-sequences-of","repo_url":"https://github.com/lilj999/MonotonicCollatz","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}