Formulation Lonely runner conjecture

consider k runners on circular track of unit length. @ t = 0, runners @ same position , start run; runners speeds pairwise distinct. runner said lonely @ time t if @ distance of @ least 1/k every other runner @ time t. lonely runner conjecture states each runner lonely @ time.

a convenient reformulation of conjecture assume runners have integer speeds, not divisible same prime; runner lonely has 0 speed. conjecture states set d of k − 1 positive integers greatest common divisor 1,

∃
t
∈

r


∀
d
∈
d

∥
t
d
∥
≥


1
k


,


{\displaystyle \exists t\in \mathbb {r} \quad \forall d\in d\quad \|td\|\geq {\frac {1}{k}},}

where ||x|| denotes distance of real number x nearest integer.