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The Lonely Runner Cojecture
Jason Holman
Areas
 Number Theory
 Diophantine Equations
 Graph Theory
 Open questions in this area
 Traces of combinatorics
 Logic
History of the Problem
 JorgWills first discovered the problem in 1967
 Thomas Cusick found it independently
 Given a name by Luis Goddyn
What is it?
 In a mathematical sense, it is the following equation
 In a general sense, it says that if “runners” are running on a
track of unit length at distinct speeds, every runner will at
some point be
from all other runners at some point
 http://en.wikipedia.org/wiki/Lonely_runner_conjecture
 This is something that seems to be quite obvious, yet shows
to be very difficult to prove
Proofs of Cases So Far
 k=1 is a trivial case
 k=2 is a trivial case
 k=3 is said to be included in all cases greater than 3
 k=4 was proven in the 1970’s by Betke and Wills
 k=5 was proven in the 1980’s by Cusick and Pomerance.
This required computer checking
 Bienia and others gave a simpler proof for this case in the 1990’s
 k=6 was proven by Bohman, Holzman, and Kleitman in 2001
 A simpler proof of this case was given by Renault in 2004
 k=7 was proven in 2008 by Barajas and Serra
Open Problem
 The conjecture has been proven for cases up to k=7
 Cases where k is greater than 7 or a general proof have not
yet been found
 There does not appear to be a certain way to “attack” this
proof
 PDF of proofs
 3 and 5 five runners have similar proofs
 The rest are quite different and very in depth
Sources
 http://blogs.ams.org/mathgradblog/2013/08/22/lonely-
runner-conjecture/
 http://rjlipton.wordpress.com/2012/01/28/the-lonelyrunner-conjecture/
 http://stathletics.tumblr.com/post/21662762724/thelonely-runner-conjecture
 Barajas, Serra. The lonely runner with seven runners. 2008