Eleven, twelve, and thirteen lonely runners
Advances the proof of a long-standing combinatorial number theory conjecture from k=9 to k=12, providing incremental progress for the mathematics community.
The authors prove the Lonely Runner Conjecture for up to 12 runners (k=10,11,12) using a refined computational method with new sieving techniques and a polynomial method argument.
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 $k\in\{10,11,12\}$.