Nine and ten lonely runners

TL;DR

This paper extends computer-assisted proofs of the Lonely Runner Conjecture to 9 and 10 runners by refining previous methods with a sieve technique.

Contribution

It improves existing proofs of the conjecture for more runners using a refined computational approach.

Findings

Proofs for 9 and 10 runners obtained via computer-assisted methods.

Refinement of previous approaches with a sieve technique.

Extension of the conjecture's verified cases to higher numbers of runners.

Abstract

The Lonely Runner Conjecture of Wills and Cusick states that if $k+1$ runners start running at distinct constant speeds around a unit-length circular track, then for each runner there is a time when he/she is at least $1/(k+1)$ away from all other runners. Rosenfeld recently obtained a computer-assisted proof of the conjecture for $8$ runners. By refining his approach with a sieve, we obtain proofs (also computer-assisted) for $9$ and $10$ runners.