The lonely runner conjecture in 2026: proved up to 13 runners (and claims for 15)
After decades stuck at 7 runners, computer-assisted proofs settled 8 to 13 runners in 2025–2026, and a September 2026 preprint claims 14 and 15. What the conjecture says, how the new proofs work, and what is left to check.
Updated 2026-10-03 · CC BY 4.0
Picture k + 1 runners on a circular track of length 1, all starting together, each running at a different constant speed. The lonely runner conjecture says that each runner is, at some moment, at distance at least 1/(k + 1) from all the others — "lonely". It was posed by J. M. Wills in 1967 in the language of Diophantine approximation, and independently by T. W. Cusick in 1974 as a view-obstruction problem; the picture of runners on a track was first published in 1998. For more than fifteen years it was stuck at seven runners. Since 2025 it has moved faster than in the previous four decades.
The statement
By fixing one runner and looking at the others from its point of view, the conjecture becomes a statement about k non-zero integer speeds u_1, …, u_k:
There is a real time t such that ‖t·u_i‖ ≥ 1/(k + 1) for every i,
where ‖x‖ is the distance from x to the nearest integer. The bound 1/(k + 1) cannot be improved: for speeds 1, 2, …, k it is attained exactly. Counting conventions differ — some papers count the speeds (k), others the runners (k + 1). This guide counts runners.
Status as of October 2026
| Runners | Proved by | Year |
|---|---|---|
| up to 4 | Betke and Wills | 1972 |
| 5 | Cusick and Pomerance | 1984 |
| 6 | Bohman, Holzman and Kleitman (shorter proof: Renault, 2004) | 2001 |
| 7 | Barajas and Serra | 2008 |
| 8 | Rosenfeld | 2025 |
| 9 | Rosenfeld; independently Trakulthongchai | 2025 |
| 10 | Trakulthongchai | 2025 |
| 11–13 | Sungkawichai and Trakulthongchai | 2026 |
| 14–15 | Allikvee (claimed, not yet independently checked) | 2026 |
The proofs for 8 runners and beyond are preprints (arXiv 2509.14111, arXiv 2511.22427, arXiv 2604.23906); the 14–15 claim is arXiv 2609.02604 from September 2026. None has yet appeared in a journal, which for computer-assisted proofs usually means the computation has not been independently re-run.
How the new proofs work
All of the recent results follow the same two-step pattern.
1. Reduce to finitely many cases. If the conjecture fails for some number of runners, there is a counterexample with bounded speeds. Terence Tao showed in 2018 that it suffices to check speeds up to a bound depending only on the number of runners; later work sharpened such bounds and added structural constraints on a minimal counterexample (for instance, divisibility conditions on the speeds). Each improvement shrinks the set of speed tuples that must be examined.
2. Check every remaining case by computer. For each tuple of speeds left after the reduction, find a time t at which the lonely condition holds — or show that a covering argument rules the tuple out. A witness time is a rational number, so each case can be checked in exact arithmetic.
The art is in step 1: the naive bound is far too large to search, and every additional runner multiplies the work. That is why progress came in a burst once better reductions and faster search code arrived.
What is left to do
The conjecture itself is open for every number of runners above the verified range, and no approach in sight handles all of them at once. But there is concrete, checkable work now:
- Independent verification of the 8–13 runner proofs, and especially of the 14–15 claim. Re-implementing the search from the paper — not re-running the authors' code — is the strongest check.
- Extending the range to 16 runners and beyond, with released code, logs and certificates.
- Sharper reductions: better bounds on the speeds of a minimal counterexample.
- Formalisation: the reduction lemmas and small cases are well suited to Lean; a kernel-checked proof for 8 runners would be a milestone.
- Documented dead ends: sieve or covering strategies that blow up, with measured growth, so others know what not to try.
How results are checked here
The lonely runner problem page asks for computational cases in a form anyone can verify: the reduction theorem used (with proof or citation), the program, and a certificate listing, for every speed tuple left after the reduction, a rational time t that makes the stationary runner lonely (or the covering argument used). A short exact-arithmetic script re-checks every witness; reviewers check that the reduction is complete. New lemmas go through review. The page also carries a Lean statement of the full conjecture as a target.
Further reading
- Lonely runner conjecture on Wikipedia, with the full history and references.
- M. Rosenfeld, The lonely runner conjecture holds for eight runners (2025).
- T. Trakulthongchai, Nine and ten lonely runners (2025).
- P. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners (2026).