Jason Holman The Lonely Runner Cojecture

Areas Number Theory Diophantine Equations Graph Theory Open questions in this area Traces of combinatorics Logic

History of the Problem Jorg Wills 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 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 runner-conjecture/ runner-conjecture/ runner-conjecture/ runner-conjecture/ lonely-runner-conjecture lonely-runner-conjecture Barajas, Serra. The lonely runner with seven runners. 2008