[#P33] Lonely runner conjecture

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.

Runners on a circular track with separation arcs.

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Definitions and notation

1Context

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

2Definitions

Definition 1 (Circular distance). Circular distance is the length of the shorter arc between two positions on a circle of circumference 1.

Definition 2 (A runner). A runner is lonely at a time if its circular distance from every other runner is at least 1/n.

3What 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.

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: 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][2][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	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: 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][2][3] Relevance to this problem For Lonely runner 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 1, Reference 2, Reference 3 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Lonely runner conjecture, record lonely-runner-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. 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 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. 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 2024 survey describes the conjecture as widely open. 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 survey records equivalent formulations and solved cases. Check those results before presenting a dimension-specific argument as new.

Computational notes

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

2See also

Closest prime square to the cube of a prime below one trilliondiophantine approximation
Unbounded continued-fraction coefficients of pidiophantine approximation
Littlewood conjecturediophantine approximation

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Plain text

“Lonely runner conjecture.” TheoremDB. P33. Problem statement; statement text SHA-256 e33611e97960c2b2ed5bc29a046cabaddb90968609531194c4d004e0f2b30952. https://theoremdb.org/statement/?ref=P33

BibTeX

@misc{theoremdb-problem-e33611e97960c2b2ed5bc29a046cabaddb90968609531194c4d004e0f2b30952,
  title = {{Lonely runner conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e33611e97960c2b2ed5bc29a046cabaddb90968609531194c4d004e0f2b30952},
  url = {https://theoremdb.org/statement/?ref=P33}
}

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. Guillem Perarnau and Oriol Serra, “The Lonely Runner Conjecture turns 60”. arXiv:2409.20160 (2024). Guillem Perarnau and Oriol Serra, survey, arXiv:2409.20160. ↗preprint · primary source · arXiv:2409.20160, checked 2026-07-31 · checked 2026-07-31Source 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.Also cited at Survey status discussion.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.Packet-linked survey and historical baseline.Source named by the research packet.
Matthieu Rosenfeld, “The lonely runner conjecture holds for eight runners”. arXiv:2509.14111 (2025). Abstract and theorem. ↗preprint · primary source · arXiv:2509.14111v2 · checked 2026-08-01Source use: original summary.Proves the eight-runner case.
Touch Sungkawichai and Tanupat Trakulthongchai, “Eleven, twelve, and thirteen lonely runners”. arXiv:2604.23906 (2026). Abstract and convention statement. ↗preprint · primary source · arXiv:2604.23906v1 · checked 2026-08-01Source use: original summary.Computer-assisted verification through 13 total runners.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P33] Lonely runner conjecture

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3What counts as a solution

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Computational notes

2See also

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Computational notes

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Background

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