{
  "abstract": "We prove the Lonely Runner Conjecture for fourteen and fifteen runners. Our proof combines a stronger bound on the speed product in a primitive counterexample with exhaustive computations modulo primes. To obtain the bound, we work with a projected lattice basis and use cases of the conjecture with fewer runners to bound partial sums of its squared Gram-Schmidt lengths. For fifteen runners, the product bound derived from the work of Malikiosis, Santos, and Schymura gives a logarithmic threshold of about $810$; ours reduces this to about $415$, making the computation feasible. For fourteen runners, the new bound reduces the required number of primes from $111$ to $61$. The computations start with a complete two-branch covering search, followed by binary lifting. Since $14$ and $15$ are composite, the polynomial argument used in earlier work does not apply directly. We finish the fourteen-runner case by a direct search. For fifteen runners, we use the factorisation $15=3\\cdot5$ and a shifting argument when all but a few speeds share a common divisor. The code and certificates are publicly archived.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02604",
  "author_total": 1,
  "authors": [
    {
      "name": "Jaan Allikvere"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d55q4ntnw44q1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02604",
  "fulltext_url": null,
  "id": "d55q4ntnw44q1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02604",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7207685584",
  "pdf_url": "https://arxiv.org/pdf/2609.02604",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d55q4ntnw44q1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.02604",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02604"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fourteen and fifteen lonely runners",
  "title_text": "Fourteen and fifteen lonely runners",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/d55q4ntnw44q1",
  "venue": "arXiv",
  "versions": []
}