Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The lonely runner conjecture holds for eight runners

Matthieu Rosenfeld

TL;DR

This work proves the lonely runner conjecture for eight runners by coupling a computable bound on the product of speeds in a minimal counterexample with a prime-divisibility framework and substantial computer verification. The argument leverages Malikiosis–Santos–Schymura reductions to derive a tight product bound and then forces divisibility by many primes, yielding a contradiction. The same strategy yields explicit results for the known cases with 4–7 runners and points to a feasible path for addressing 9–10 runners with further computational and methodological advances. Additionally, the paper strengthens lacunary-sequence results, reducing the required gaps and broadening the scope of LR-type guarantees in diophantine settings.

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.

The lonely runner conjecture holds for eight runners

TL;DR

This work proves the lonely runner conjecture for eight runners by coupling a computable bound on the product of speeds in a minimal counterexample with a prime-divisibility framework and substantial computer verification. The argument leverages Malikiosis–Santos–Schymura reductions to derive a tight product bound and then forces divisibility by many primes, yielding a contradiction. The same strategy yields explicit results for the known cases with 4–7 runners and points to a feasible path for addressing 9–10 runners with further computational and methodological advances. Additionally, the paper strengthens lacunary-sequence results, reducing the required gaps and broadening the scope of LR-type guarantees in diophantine settings.

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.

Paper Structure

This paper contains 8 sections, 10 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Upper bound on the product of the speeds in a minimal counterexample
  3. Finding a prime that divides the product of the speeds
  4. Proof of the lonely runner conjecture for eight runners
  5. The cases of 4, 5, 6 and 7 runners
  6. Implementation details
  7. Possible further improvement
  8. Improving the lacunary case

Key Result

Theorem 1

For all set of integers $\{v_1,\ldots, v_7\}$ of size $7$ there exists a real number $t$ such that for all $i$,

Theorems & Definitions (17)

  • Conjecture 1: Lonely Runner Conjecture
  • Conjecture 2: Lonely Runner Conjecture
  • Theorem 1
  • Theorem 2: Malikiosis2024Nov
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 7 more