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Jason Holman The Lonely Runner Cojecture

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

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History of the Problem Jorg Wills first discovered the problem in 1967 Thomas Cusick found it independently Given a name by Luis Goddyn

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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

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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

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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

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Sources http://blogs.ams.org/mathgradblog/2013/08/22/lonely- runner-conjecture/ http://blogs.ams.org/mathgradblog/2013/08/22/lonely- runner-conjecture/ http://rjlipton.wordpress.com/2012/01/28/the-lonely- runner-conjecture/ http://rjlipton.wordpress.com/2012/01/28/the-lonely- runner-conjecture/ http://stathletics.tumblr.com/post/21662762724/the- lonely-runner-conjecture http://stathletics.tumblr.com/post/21662762724/the- lonely-runner-conjecture Barajas, Serra. The lonely runner with seven runners. 2008

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