The conjecture
Place k runners on a circular track of unit length, all at the starting line. Send them off at distinct integer speeds. Run forever. The lonely runner conjecture says: for each runner there's some moment when no other runner is within 1/k of them. Equivalently — and the equivalence is by passing to the frame of the runner you're observing — for any k−1 distinct nonzero integers v₁, …, vk−1, there exists a time t such that the fractional parts {vi·t} all lie in [1/k, 1 − 1/k].
Joseph Wills posed it in 1967; Tom Cusick gave it the name "lonely runner" in 1973 while studying view-obstruction problems in the geometry of numbers.
What's hard about it
For small k the conjecture is almost obvious — fewer runners means more places for them to spread out, and the lonely interval (1/k, 1 − 1/k) is wide. As k grows the lonely interval shrinks to a sliver near the back of the track, and you need more runners to simultaneously be in that sliver. The Diophantine problem of when v₁t, …, vk−1t can simultaneously avoid a small neighbourhood of zero becomes delicate.
History
- 1967 — Wills poses the question.
- 1973 — Cusick reframes it as a view-obstruction problem and gives it the "lonely runner" name. Proves k = 4.
- 1984 — Cusick & Pomerance prove k = 5.
- 2001 — Bohman, Holzman, & Kleitman prove k = 6 using a careful case analysis.
- 2008 — Barajas & Serra prove k = 7.
- k ≥ 8 — open. Believed true.
Equivalent forms
The lonely runner problem is also a question about covering integers by arithmetic progressions, a question about view-obstruction in the geometry of numbers (asking when integer-point lines from the origin in ℝⁿ are "seen" by a small ball), and a question about flow on a torus (the runner positions are a flow line on the n-torus, and the conjecture asks about avoiding a thickening of a coordinate hyperplane). Different fields prove different cases using their own tools.
Why this matters beyond a track
The conjecture is a clean test case for our understanding of simultaneous Diophantine approximation — how independently can k linear forms over the integers behave modulo 1? Every time someone proves a new k, the proof unlocks a related family of problems. Bohman–Holzman–Kleitman's k = 6 proof in particular introduced combinatorial-Fourier tools that have been useful elsewhere.
What's in this site
- track — the canonical animation. k runners around a circle. The avoidance arc (the 1/k-neighbourhood of the lonely runner's position) is shaded; runners outside it count as "lonely-from-this-runner". An all-lonely moment lights the centre. The slider lets you pick k, the speed presets vary which integer speeds the other runners use, and the find lonely button does a forward search until it finds a simultaneous all-lonely time.
- docs — this page.
Why this sits alongside erdős, guthkatz, and hadwiger
The geometry pack so far: erdős (extremal point counts in the plane), guthkatz (extremal distance counts), hadwiger (colouring the plane). runner is the first that's not strictly in the Euclidean plane — it's on the circle, which is a torus. But the family resemblance is real: an extremal-combinatorics question with a small-integer answer, a long climb of partial results, and an open endpoint. The story-shape carries.
- Wills, J. M. (1967). Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen. Monatshefte für Mathematik 71, 263–269.
- Cusick, T. W. (1973). View-obstruction problems. Aequationes Mathematicae 9, 165–170.
- Bohman, T., Holzman, R. & Kleitman, D. (2001). Six lonely runners. Electronic Journal of Combinatorics 8.
- Barajas, J. & Serra, O. (2008). The lonely runner with seven runners. Electronic Journal of Combinatorics 15.
- Wikipedia — Lonely runner conjecture