The lonely runner conjecture is a mathematical hypothesis which states that, for n points ("runners") moving around a circle at distinct constant speeds, each runner will always be "lonely" at some time - that is, there is some time when that runner will be a distance of at least 1/n from all other runners.

Equivelently, for any distinct constants v₁, v₂, ..., vₙ₋₁, there is some t such that 1/n ≤ {vᵢt} ≤ 1 - 1/n for all 1 ≤ i ≤ n-1, where {•} denotes the fractional part. (This equivalence comes from subtracting one speed from all the others, so that the runner with that speed is now stationary, and the other runners' distances from it are given by min{{vt}, {1-vt}}).

The original conjecture allows for negative speeds, but it is equivalent to the case where all speeds are positive, since you can add a constant to all of the speeds without changing whether any of them will become lonely. The second formulation can also be restricted to positive speeds.

See more about the conjecture:

https://en.wikipedia.org/wiki/Lonely_runner_conjecture

https://www.cantorsparadise.com/the-lonely-runner-conjecture-409b9aac4a22