十四位孤独的赛跑者
Fourteen and fifteen lonely runners
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中文总结 AI 辅助
本文通过计算机辅助扩展Sungkawichai等人的有限验证框架，结合归纳输入、素数门验证等方法，证明了十四位赛跑者的孤独赛跑者猜想。
中文摘要 AI 辅助
我们通过对Sungkawichai和Trakulthongchai的有限验证框架进行计算机辅助扩展，证明了十四位赛跑者的孤独赛跑者猜想。当一位赛跑者静止时，他们的十三位赛跑者结果提供了归纳输入，且他们的归约留下了由素数索引的有限个模运算。我们验证了111个此类素数门，满足∑ₚ log p > 681.5292，超过所需阈值log B₁₃ < 670.3498的部分超过11.17。对于每个门，穷尽生成器构造出一级非真族，一系列精确的二元提升过滤器消除了除两个乘法轨道外的所有轨道，且精确的分支定界计算在混合14级处理每个剩余的7¹³提升纤维。在该级剩余的所有无见证完成项的所有坐标均被7整除，因此根据框架定义中的gcd条款是真的。相同的两个持久轨道出现在每个闭合门处；这是一个经验普适性发现，而非超出已验证门集的定理。每个门的证书以及对所有111个闭合门的单独审计支持该计算。我们还报告了所选流水线未能闭合的每个门。
英文摘要
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.
