{"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/the-collatz-tree-is-a-hilbert-hotel-a-proof","title":"The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture","arxiv_id":"2008.13643","date":"2020-08-31","proceeding":null,"authors":["Jan Kleinnijenhuis","Alissa M. Kleinnijenhuis","Mustafa G. Aydogan"],"abstract":"The Collatz conjecture states that repeated steps of $n\\mathrm{\\to }\\mathrm{3}n\\mathrm{+1}$ at odd numbers and $n\\mathrm{\\to }n\\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial' cyclic root $4\\to 2\\to 1\\to 4\\to \\dots $ of one connected Collatz graph. The Collatz graph with reverse arrows $n \\to 2n$ and $n \\to (n-1)/3$ can be transformed to a 3-regular automorphic Cayley color graph $T_{\\ge 0}$ with as nodes the branching numbers with a remainder of $4$ or $16$ when divided by $18$, building the congruence classes $[4,16]_{18}$. Labeling the $2^k$ breadth-first ordered root paths with $2^k$ binary numbers on the binary number line, for $k=1,2,3,\\dots$, and pairing them with the $2^k$ output numbers of these root paths, gives $2^k$ paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes $[4,16]_{18}$ a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's $3n+1$ conjecture. Whether a specific $an+b$ conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than $2a^3$.","url_abs":"https://arxiv.org/abs/2008.13643v8","url_pdf":"https://arxiv.org/pdf/2008.13643v8.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":"the-collatz-tree-is-a-hilbert-hotel-a-proof","repo_url":"https://github.com/c4ristian/collatz","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}