[#P19] Collatz conjecture
Problem. Define \(T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}\) by \(T(n)=n/2\) when \(n\) is even and \(T(n)=3n+1\) when \(n\) is odd. For every \(n\ge 1\), there exists \(k\ge 0\) such that \(T^k(n)=1\).
1Context
The update rule is elementary to compute, while the long-term behavior of all positive-integer orbits remains unresolved.
2Problem setup
Definition 1 (A Collatz orbit). A Collatz orbit is the sequence obtained by repeatedly applying the even-or-odd update rule.
Definition 2 (Reaching 1). Reaching 1 is equivalent to entering the cycle 1, 4, 2, 1 under this rule.
Remark 1. The update rule is elementary to compute, while the long-term behavior of all positive-integer orbits remains unresolved.
3What counts as a solution
- Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Tao proves that almost all Collatz orbits, in logarithmic density, attain any prescribed bound tending to infinity. This does not prove that every positive-integer orbit reaches 1. Exact unresolved remainder: Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1.[2][1]
1Records
Notes and companion material
Original intake status. The cited scholarly overview treats this as an unsolved problem. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- The source surveys equivalent formulations, verified ranges, and partial results; consult it before claiming novelty.
Recorded example 1. Starting from 6 gives 6, 3, 10, 5, 16, 8, 4, 2, 1.
Computational notes
- Verification through any finite bound leaves infinitely many starting values untreated.
2See also
- Longest rotor-router cover time on the eight by eight griddiscrete dynamical systems
- Longest four-number Ducci trajectory in the 100 boxdiscrete dynamical systems
- Merging of orbits under adding the product of nonzero digitsdiscrete dynamical systems
How to cite
TheoremDB contributors, “Collatz conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/collatz-conjectureThis page as plain text: collatz-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Jeffrey C. Lagarias, “The 3x+1 Problem: An Overview”. The Ultimate Challenge: The 3x+1 Problem, Edited by Jeffrey C. Lagarias. American Mathematical Society, Providence RI 2010, pp. 3--29. arXiv:2111.02635 (2021). Jeffrey C. Lagarias, survey overview, arXiv:2111.02635. ↗preprint · primary source · arXiv:2111.02635, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited scholarly overview treats this as an unsolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at overview and open-problem formulation.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides the standard formulation and status context.Source named by the research packet.
- Terence Tao, “Almost all orbits of the Collatz map attain almost bounded values”. Forum Math. Pi 10 (2022), Paper No. e12, 56 pp. arXiv:1909.03562 (2019). abstract and main theorem. ↗preprint · primary source · arXiv:1909.03562v7 · checked 2026-08-01Source use: original summary.Provides the strongest checked almost-all orbit result while leaving the universal conjecture open.
An original CC0 restatement prepared by TheoremDB maintainers.