[#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 \(T^k(n)=1\).

A finite initial segment of the exact Collatz trajectory starting at 27.

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Definitions and notation

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.

Remark 1. Reaching 1 is equivalent to entering the cycle 1, 4, 2, 1 under this rule.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Collatz conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1.^[1] Relevance to this problem For Collatz conjecture, record collatz-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Merging of orbits under adding the product of nonzero digitsdiscrete dynamical systems
Integer return under the 4/3 ceiling mapdiscrete dynamical systems
Trapping the origin-first three-dimensional knight walkdiscrete dynamical systems

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Plain text

“Collatz conjecture.” TheoremDB. P19. Problem statement; statement text SHA-256 e2ef8ecb05a7e3bfeb499aadb5dc9bbf61be0c20d9cc338ae7c66e6da16d63d9. https://theoremdb.org/statement/?ref=P19

BibTeX

@misc{theoremdb-problem-e2ef8ecb05a7e3bfeb499aadb5dc9bbf61be0c20d9cc338ae7c66e6da16d63d9,
  title = {{Collatz conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e2ef8ecb05a7e3bfeb499aadb5dc9bbf61be0c20d9cc338ae7c66e6da16d63d9},
  url = {https://theoremdb.org/statement/?ref=P19}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P19] Collatz conjecture

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3What counts as a solution

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