Solving Lonely Runner Conjecture through differential geometry

Abstract

Abstract The Lonely Runner Conjecture is a known open problem that was defined by Wills in 1967 and in 1973 also by Cusick independently of Wills. If we suppose n runners having distinct constant speeds start at a common point and run laps on a circular track with a unit length, then for any given runner, there is a time at which the distance of that runner is at least 1/n from every other runner. There exist several hypothesis verifications for different n mostly based on principles of approximation using number theory. However, the general solution of the conjecture for any n is still an open problem. In our work we will use a unique approach to verify the Lonely Runner Conjecture by the methods of differential geometry, which presents a non-standard solution, but demonstrates to be a suitable method for solving this type of problems. In the paper we will show also the procedure to build an algorithm that shows the possible existence of a solution for any number of runners.