Open problem, Combinatorics and discrete mathematics, Arithmetic and number theory, posed 1967

Lonely runner conjecture

Open

Consider $n$ runners on a circular track of circumference 1, all starting from the same point at time 0 and each running at a distinct constant speed. A runner is called lonely at time $t$ if the circular distance from that runner to every other runner is at least $\frac{1}{n}$ . The conjecture asserts that every runner is lonely at some time $t > 0$ .

Research frontier as of 2026

As of 2026 the lonely runner conjecture is open for general $n$ . A remarkable series of computer-assisted proofs in 2025–2026 extended the verified range from 7 runners (Barajas–Serra, 2008) up to 13 runners. The key breakthroughs are preprints by Matthieu Rosenfeld (8 runners, arXiv:2509.14111; 9 runners, arXiv:2512.01912), Tanupat Trakulthongchai (9–10 runners, arXiv:2511.22427), and Sungkawichai–Trakulthongchai (11–13 runners, arXiv:2604.23906). All proofs are computer-assisted and rely on sieve techniques. No unconditional proof for general $n$ is known.

Best known results

Verified by computer-assisted proof for all $n \le 13$ runners (2026).
For speeds forming an arithmetic progression, or more generally for speeds in a Bohr set of bounded density, the conjecture is known to hold.

Tools and where they stop

Tool	Achieved	Where it stops
Computer-assisted sieve / exhaustive case analysis	Proves the conjecture for specific small values of $n$ by exhaustive verification of all speed configurations up to symmetry.	The search space grows super-exponentially with $n$ , making this approach infeasible for large $n$ without a fundamentally new structural insight.
Bohr sets and additive combinatorics	Establishes weaker density-type results: the conjecture holds when the speeds lie in structured arithmetic sets of bounded density.	Cannot force the conjecture for arbitrary distinct integer speeds without additional structure.

Open questions

Is there a unified algebraic or analytic proof strategy that handles all $n$ without case-by-case machine verification?
Does the conjecture hold when runners are allowed to have real (non-integer) speeds, or can a counterexample be constructed for non-integer speeds?

References

Jordi Barajas, Oriol Serra (2008). The lonely runner with seven runners · DOI:10.37236/772
Matthieu Rosenfeld (2025). The lonely runner conjecture holds for eight runners · arXiv:2509.14111 [preprint, not peer-reviewed]
Tanupat Trakulthongchai (2025). Nine and ten lonely runners · arXiv:2511.22427 [preprint, not peer-reviewed]
Touch Sungkawichai, Tanupat Trakulthongchai (2026). Eleven, twelve, and thirteen lonely runners · arXiv:2604.23906 [preprint, not peer-reviewed]
Bela Bajnok (2007). The Lonely Runner Problem