The Lonely Runner Conjecture - Proven or Not!!

Hello All,

There is an intriguing problem called the Lonely Runner Conjecture. Its intriguing because it is easily stated yet deceptively difficult to prove. The conjecture has been open for more than 45 years and it has important implications across many areas of mathematics. Let me share the conjecture:

There is a circular running track of unit length upon which n runners, run at different constant **integer** speeds (they run forever). The conjecture says that during the race all runners will at some point become lonely - meaning no closer than 1/n to any other runners.

Here is my question to you: is **(2+***∆ *) where *∆ **→ ** 0* close enough to 2 (could be any integer) that for this problem it doesn't matter! If so then there is a link to a proof below which is probably correct. If the proof isn't correct its only *∆* away. The proof is novel but its not mathematically that challenging, so have a look! It might inspire you!

http://vixra.org/pdf/1210.0140v2.pdf

(you might need to copy and paste this into your browser)

Comments would be great!

Pat