The lonely runner conjecture

Runners with distinct constant speeds start together on a circular track. The lonely runner conjecture says that each of them is, at some moment, far from all the others. Computer-assisted proofs since 2025 have pushed it to 15 runners, but no general proof is known, and a counterexample would be a single list of integer speeds.

Problem

For a real number $x$, let $\|x\|$ denote its distance to the nearest integer.

Conjecture. For every $n \ge 1$ and all distinct positive integers $v_1, \dots, v_n$, there is a real number $t$ with

$$\|t v_i\| \ge \frac{1}{n+1} \quad \text{for all } i = 1, \dots, n.$$

In runner language, a stationary runner sits at $0$ on a track of length 1 while $n$ runners move with speeds $v_1, \dots, v_n$; at time $t$ the stationary runner is lonely if every other runner is at least $\frac{1}{n+1}$ away.

Task. Find distinct positive integers $v_1, \dots, v_n$ for which no such $t$ exists, or prove the conjecture.

Background

The statement goes back to Wills (1967), in Diophantine approximation [1], and to Cusick (1973), in view-obstruction problems. Bienia, Goddyn, Gvozdjak, Sebő and Tarsi connected it with nowhere-zero flows in graphs under the name "lonely runner" [2]. Proofs for small $n$ accumulated slowly: Barajas and Serra handled seven runners ($n = 6$) in 2008. Nearly two decades later, Rosenfeld proved the case of eight runners with a computer-assisted method [3], Trakulthongchai, with Sungkawichai, extended it to nine through thirteen runners [4, 5], and Allikvere to fourteen and fifteen [7].

What is known

• The conjecture holds for $n \le 14$, that is, up to 15 runners. The cases $7 \le n \le 14$ were all proved in 2025–2026 with computer assistance [3–5, 7]; the last two, $n = 13$ and $n = 14$, are in an unrefereed September 2026 preprint [7].
• Two strengthenings have been refuted: Blanco, Criado and Santos disproved the shifted lonely runner conjecture in 2026 [6], and Fan and Sun disproved Kravitz's loneliness spectrum conjecture for $n = 4$ (posted 2023, EJC 2026). Epoch reads this as a sign that the conjecture itself could fail.
• formal-conjectures states the conjecture in Lean as lonely_runner_conjecture in LonelyRunnerConjecture.lean.

What would count as progress

• A list of speeds. Checking it is exact: for integer speeds the condition is periodic in $t$ with period 1, and for each $i$ the set of $t \in [0, 1)$ with $\|t v_i\| < \frac{1}{n+1}$ is a finite union of intervals, so one checks whether these intervals cover $[0, 1)$.
• Epoch scores submissions with its own verifier, which takes the list of speeds and confirms that no lonely time exists.
• A counterexample with visible arithmetic structure would be the most informative outcome; a proof for $n = 15$ or for all $n$ would settle the other side.

References

1. J. M. Wills, Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen, Monatsh. Math. 71 (1967), 263–269. doi:10.1007/BF01298332
2. W. Bienia, L. Goddyn, P. Gvozdjak, A. Sebő and M. Tarsi, Flows, view obstructions, and the lonely runner, J. Combin. Theory Ser. B 72 (1998), 1–9. doi:10.1006/jctb.1997.1770
3. M. Rosenfeld, The lonely runner conjecture holds for eight runners, 2025. arXiv:2509.14111
4. T. Trakulthongchai, Nine and ten lonely runners, 2025. arXiv:2511.22427
5. T. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners, 2026. arXiv:2604.23906
6. M. Blanco, F. Criado and F. Santos, Coloopless zonotopes and counterexamples to the shifted lonely runner conjecture, 2026. arXiv:2603.24784
7. J. Allikvere, Fourteen and fifteen lonely runners, 2026. arXiv:2609.02604

Source

Imported from FrontierMath: Open Problems, lonely-runner, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.