My first attempt at solving the Lonely Runner Conjecture

-August 31, 2013August 31, 2013Uncategorized

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 $n\in\Bbb{Z}^+$