# P19: Collatz conjecture

- ID: `P19`
- Reference: `collatz-conjecture`
- Page: https://theoremdb.org/statements/P19
- Record maturity: Reviewed problem with recorded work

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

### Context

The update rule is elementary to compute, while the long-term behavior of all positive-integer orbits remains unresolved.

### Problem setup

- **Definition (A Collatz orbit).** A Collatz orbit is the sequence obtained by repeatedly applying the even-or-odd update rule.
- **Definition (Reaching 1).** Reaching 1 is equivalent to entering the cycle 1, 4, 2, 1 under this rule.
- **Remark.** The update rule is elementary to compute, while the long-term behavior of all positive-integer orbits remains unresolved.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- The statement and status were checked against Lagarias's overview on 2026-07-22.
- The source surveys equivalent formulations, verified ranges, and partial results; consult it before claiming novelty.

- Recorded example: Starting from 6 gives 6, 3, 10, 5, 16, 8, 4, 2, 1.

### Open directions

- **Route 1** (reported): Prove that every positive-integer orbit reaches 1, or give a positive integer whose orbit is rigorously shown never to reach 1. [1](#reference-1)

### Computational notes

- Verification through any finite bound leaves infinitely many starting values untreated.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `collatz-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2111.02635
   - Also cited at overview and open-problem formulation
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2111.02635, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/1909.03562
   - preprint; primary source; arXiv:1909.03562v7; checked 2026-08-01
   - Source use: original_summary
   - Provides the strongest checked almost-all orbit result while leaving the universal conjecture open.