Math Conjectures

Lonely Runner Conjecture

OPENMajorConjectureProposed 1967 · Standard version

Canonical statement

For every integer n≥2n\geq2 and every set of distinct positive integers v1,…,vn−1v_1,\ldots,v_{n-1}, there exists t∈Rt\in\mathbb R such that
∥tvi∥R/Z≥1n(1≤i≤n−1), \lVert tv_i\rVert_{\mathbb R/\mathbb Z}\geq\frac1n \qquad(1\leq i\leq n-1),
where ∥x∥R/Z=min⁡z∈Z∣x−z∣\lVert x\rVert_{\mathbb R/\mathbb Z} =\min_{z\in\mathbb Z}|x-z|.
View source LaTeX
For every integer \(n\geq2\) and every set of distinct
positive integers \(v_1,\ldots,v_{n-1}\), there exists \(t\in\mathbb R\)
such that
\[
  \lVert tv_i\rVert_{\mathbb R/\mathbb Z}\geq\frac1n
  \qquad(1\leq i\leq n-1),
\]
where \(\lVert x\rVert_{\mathbb R/\mathbb Z}
=\min_{z\in\mathbb Z}|x-z|\).
COMB-004:[Wills1967Lonely][PerarnauSerra2025]
CombinatoricsNumber theory

Notes

Given distinct positive integer speeds v1,…,vn−1v_1,\ldots,v_{n-1}, the conjecture asserts that some time tt exists at which every tvitv_i lies at distance at least 1/n1/n from the nearest integer. In the usual picture, nn runners start together on a circular track of unit length and run at pairwise distinct constant speeds; after subtracting one runner's speed and rescaling, the statement says that each runner is at some moment lonely, with all the others at circular distance at least 1/n1/n. The problem originates with Wills in 1967, in the setting of inhomogeneous Diophantine approximation [Wills1967Lonely].

A simple averaging argument guarantees a time when all runners are at distance on the order of 1/(2(n−1))1/(2(n-1)), so the content of the conjecture is the improvement to the sharp constant 1/n1/n. The statement has been verified for small numbers of runners and for various structured families of speed sets, and the surrounding literature is surveyed by Perarnau and Serra on the occasion of the problem's sixtieth anniversary [PerarnauSerra2025].

No method currently handles arbitrary nn together with arbitrary distinct speeds, and the conjecture remains open in general.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.