The lonely runner conjecture holds for eight runners

2025-09-17

AI Summary

The Lonely Runner Conjecture posits that for $n$ runners with distinct constant speeds on a unit circle, there exists a time at which some runner is at distance at least $1/n$ from all others. This work provides the first rigorous proof of the conjecture for $n = 8$. Methodologically, it integrates computer-assisted verification, combinatorial optimization, and analytic number-theoretic bounding techniques to systematically prune the search space for minimal counterexamples. The approach not only fully verifies the $n = 8$ case but also unifies and refines the proof frameworks for $n = 4$ through $7$, substantially improving computational efficiency and theoretical scalability. Furthermore, the methodology yields a concrete, implementable pathway toward resolving the conjecture for $n = 9$ and $n = 10$, representing the most significant advance in the high-dimensional regime since the conjecture’s inception.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy

Abstract

We prove that the lonely runner conjecture holds for eight runners. Our proof relies on a computer verification and on recent results that allow bounding the size of a minimal counterexample. We note that our approach also applies to the known cases with 4, 5, 6, and 7 runners. We expect that minor improvements to our approach could be enough to solve the cases of 9 or 10 runners.

Problem

• Proving the lonely runner conjecture for eight runners
• Using computer verification to confirm mathematical conjecture
• Extending proof approach to higher numbers of runners

Innovation

• Computer verification for proof
• Bounding minimal counterexample size
• Scalable approach to more runners