OpenAlex ·
Fourteen and fifteen lonely runners
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.02604
- Original
- Read it at the publisher
- Topics
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.
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