# P33: Lonely runner conjecture

- ID: `P33`
- Reference: `lonely-runner-conjecture`
- Page: https://theoremdb.org/statements/P33
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(n\ge 2\) runners move around \(\mathbb{R}/\mathbb{Z}\) at distinct constant speeds \(v_1,\ldots,v_n\). For each \(i\), there exists a time \(t\) such that \(\lVert t(v_j-v_i)\rVert\ge 1/n\) for every \(j\ne i\), where \(\lVert x\rVert\) is the distance from \(x\) to the nearest integer.

### Context

The problem has equivalent forms in simultaneous Diophantine approximation and view-obstruction geometry.

### Problem setup

- **Definition (Circular distance).** Circular distance is the length of the shorter arc between two positions on a circle of circumference 1.
- **Definition (A runner).** A runner is lonely at a time if its circular distance from every other runner is at least 1/n.
- **Remark.** The problem has equivalent forms in simultaneous Diophantine approximation and view-obstruction geometry.

### What counts as a solution

- Prove the separation assertion for every n and every set of distinct constant speeds, or give explicit distinct speeds and prove that at least one runner never attains the required distance from all others.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The conjecture has computer-assisted proofs through 13 total runners under the convention used in arXiv:2604.23906. Exact unresolved remainder: Prove the 1/n separation threshold for every finite number n of runners with distinct constant speeds. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## 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: The conjecture has computer-assisted proofs through 13 total runners under the convention used in arXiv:2604.23906. Exact unresolved remainder: Prove the 1/n separation threshold for every finite number n of runners with distinct constant speeds.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The conjecture has computer-assisted proofs through 13 total runners under the convention used in arXiv:2604.23906.

Exact unresolved remainder: Prove the 1/n separation threshold for every finite number n of runners with distinct constant speeds.

### Background and intake notes

- Original intake status: The cited 2024 survey describes the conjecture as widely open. 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 the cited survey on 2026-07-22.
- The survey records equivalent formulations and solved cases. Check those results before presenting a dimension-specific argument as new.

### Open directions

- **Route 1** (reported): Prove the separation assertion for every n and every set of distinct constant speeds, or give explicit distinct speeds and prove that at least one runner never attains the required distance from all others. [1](#reference-1)

### Computational notes

- Finite checking can settle a specified runner count or bounded velocity range without proving the full conjecture.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `lonely-runner-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>Guillem Perarnau and Oriol Serra, “The Lonely Runner Conjecture turns 60”. arXiv:2409.20160 (2024). Guillem Perarnau and Oriol Serra, survey, arXiv:2409.20160 https://arxiv.org/abs/2409.20160
   - Also cited at Survey status discussion
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2409.20160, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited 2024 survey describes the conjecture as widely open. 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.
   - Packet-linked survey and historical baseline.
   - Source named by the research packet.
2. <a id="reference-2"></a>Matthieu Rosenfeld, “The lonely runner conjecture holds for eight runners”. arXiv:2509.14111 (2025). Abstract and theorem https://arxiv.org/abs/2509.14111
   - preprint; primary source; arXiv:2509.14111v2; checked 2026-08-01
   - Source use: original_summary
   - Proves the eight-runner case.
3. <a id="reference-3"></a>Touch Sungkawichai and Tanupat Trakulthongchai, “Eleven, twelve, and thirteen lonely runners”. arXiv:2604.23906 (2026). Abstract and convention statement https://arxiv.org/abs/2604.23906
   - preprint; primary source; arXiv:2604.23906v1; checked 2026-08-01
   - Source use: original_summary
   - Computer-assisted verification through 13 total runners.