Lonely runner conjecture - Printable Version +- MKLab (https://mklab.gr) +-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1) +--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3) +---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13) +---- Thread: Lonely runner conjecture (/showthread.php?tid=1299)

Lonely runner conjecture - mklabgr - 07-25-2026 Lonely runner conjecture Summary The Lonely Runner Conjecture is a famous unsolved problem in number theory and combinatorics that asks whether, for any group of runners moving at distinct constant speeds around a circular track, each runner will eventually find a moment when they are at least ($1/n$) of the track away from every other runner, where (n) is the total number of runners. First proposed by Jörg Wills in 1967 and later connected to geometric view-obstruction problems, the conjecture has become one of the most intriguing open questions in modern mathematics.  Although its statement is simple and easy to visualise, proving it in full has proved remarkably difficult. The problem has deep connections to Diophantine approximation, graph theory, geometry, and combinatorics, making it important far beyond its intuitive running analogy.  The conjecture has been completely verified for up to 13 runners, with recent computer-assisted proofs extending the known cases, but the general case remains unresolved. Over the years, mathematicians have developed numerous equivalent formulations, partial results, and specialised proofs, yet a universal proof continues to elude researchers. Its combination of an elementary statement and profound mathematical depth has made it one of the best-known open problems in discrete mathematics and number theory.  ARTICLE