Fourteen and fifteen lonely runners

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=3·5 and a shifting argument when all but a few speeds share a common divisor. The code and certificates are publicly archived.

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BibTeX RIS

BibTeX

@misc{allikvere2026fourteen,
  title = {{Fourteen and fifteen lonely runners}},
  author = {Jaan Allikvere},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02604},
  eprint = {2609.02604},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02604},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Fourteen and fifteen lonely runners
AU  - Jaan Allikvere
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.02604
UR  - https://arxiv.org/abs/2609.02604
AB  - 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=3·5 and a shifting argument when all but a few speeds share a common divisor. The code and certificates are publicly archived.
ER  -

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