Eleven, twelve, and thirteen lonely runners
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908994979758080 |
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| author | Sungkawichai, Touch Trakulthongchai, Tanupat |
| author_facet | Sungkawichai, Touch Trakulthongchai, Tanupat |
| contents | Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\ldots,u_k)\equiv(1,2,\ldots,k)\pmod{p}$ with $\gcd(u_1,\ldots,u_k)=1$ satisfies the conjecture when $k+1$ and $p > k^2+k$ are both odd primes. Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for $k\in\{10,11,12\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eleven, twelve, and thirteen lonely runners Sungkawichai, Touch Trakulthongchai, Tanupat Combinatorics Discrete Mathematics Number Theory 11K60 Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\ldots,u_k)\equiv(1,2,\ldots,k)\pmod{p}$ with $\gcd(u_1,\ldots,u_k)=1$ satisfies the conjecture when $k+1$ and $p > k^2+k$ are both odd primes. Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for $k\in\{10,11,12\}$. |
| title | Eleven, twelve, and thirteen lonely runners |
| topic | Combinatorics Discrete Mathematics Number Theory 11K60 |
| url | https://arxiv.org/abs/2604.23906 |