# 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](https://doi.org/10.1007/BF01832623)), 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](https://doi.org/10.37236/772). 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](https://doi.org/10.37236/13840)). 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](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/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](https://doi.org/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](https://doi.org/10.1006/jctb.1997.1770)
3. M. Rosenfeld, The lonely runner conjecture holds for eight runners, 2025. [arXiv:2509.14111](https://arxiv.org/abs/2509.14111)
4. T. Trakulthongchai, Nine and ten lonely runners, 2025. [arXiv:2511.22427](https://arxiv.org/abs/2511.22427)
5. T. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners, 2026. [arXiv:2604.23906](https://arxiv.org/abs/2604.23906)
6. M. Blanco, F. Criado and F. Santos, Coloopless zonotopes and counterexamples to the shifted lonely runner conjecture, 2026. [arXiv:2603.24784](https://arxiv.org/abs/2603.24784)
7. J. Allikvere, Fourteen and fifteen lonely runners, 2026. [arXiv:2609.02604](https://arxiv.org/abs/2609.02604)

## Source

Imported from [FrontierMath: Open Problems, lonely-runner](https://epoch.ai/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.