My first attempt at solving the Lonely Runner Conjecture

by ayushkhaitan3437

Let us suppose there are k runners running at speeds 0<a_1<a_2<\dots<a_k around a field of circumference 1. Take any runner from the k runners- say r_i, who runs with speed a_i. Say we pair him up with another runner r_j who runs with speed a_j. Then for time 0\leq t\leq\frac{1}{2|a_i-a_j|}, the distance between them is |a_i-a_j|t, and for \frac{1}{2|a_i-a_j|}\leq t\leq \frac{1}{|a_i-a_j|}, the distance between them is 1-|a_i-a_j|t.

The Lonely Runner Conjecture can be stated in the following way: there exists a time t such that \min\left[\min\{|a_i-a_1|t_1,1-|a_i-a_1|t_1\},\min\{|a_i-a_2|t_2,1-|a_i-a_2|t_2\},\dots,\min\{|a_i-a_k|t_k,1-|a_i-a_k|t_k\}\right]\geq\frac{1}{k}.

Here t_1,t_2,t_3,\dots,t_k are the smallest positive values found after successively determining t_x-n\frac{1}{|a_i-a_x|}, where x\in\{1,2,3,\dots k\}\setminus\{i\}, and n\in\Bbb{Z}^+