<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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 <italic>n</italic> 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/<italic>n</italic> from every other runner. There exist several hypothesis verifications for different <italic>n</italic> mostly based on principles of approximation using number theory. However, the general solution of the conjecture for any <italic>n</italic> 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.</p>
</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.

Solving Lonely Runner Conjecture through differential geometry

Constantine The Philosopher University in Nitra

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Solving Lonely Runner Conjecture through differential geometry"}

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...

Solving Lonely Runner Conjecture through differential geometry

Published Online: Jul 04, 2022

Page range: 21 - 28

DOI: https://doi.org/10.2478/jamsi-2022-0002

Keywords
lonely runner problem, geometric interpretation, differential geometry, kinematics

© 2022 V. Ďuriš et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Solving Lonely Runner Conjecture through differential geometry

Published Online: Jul 04, 2022

Page range: 21 - 28

DOI: https://doi.org/10.2478/jamsi-2022-0002

Keywordslonely runner problem, geometric interpretation, differential geometry, kinematics

© 2022 V. Ďuriš et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
lonely runner problem, geometric interpretation, differential geometry, kinematics