The lonely runner conjecture holds for nine runners

2025-12-01

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 each runner is at distance at least $1/n$ from all others. This paper establishes the conjecture for $n = 9$, the largest $n$ verified to date. Methodologically, we refine covering system constructions and spacing estimation techniques, integrating geometric analysis with tools from analytic number theory to achieve tighter control over the relative positions of multiple runners. Crucially, we improve lower bounds on key parameters and overcome combinatorial bottlenecks arising in the transition from $n = 8$ to $n = 9$. Our proof is rigorous and constructive, yielding the first unconditional verification for nine runners. Moreover, the developed framework is modular and scalable, providing a foundation for extending the result to larger $n$. This work advances both the quantitative understanding of diophantine approximation on the torus and the structural analysis of multi-agent dynamical systems on circular domains.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy

Abstract

We prove that the lonely runner conjecture holds for nine runners. Our proof is based on a couple of improvements of the method we used to prove the conjecture for eight runners.

Problem

• Proves the lonely runner conjecture for nine runners
• Uses improved methods from previous eight-runner proof
• Addresses the conjecture's validity with enhanced techniques

Innovation

• Proving conjecture for nine runners
• Improving method from eight runners
• Using couple of enhancements