Six Lonely Runners

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Six Lonely Runners

For x real, let {x} be the fractional part of x (i.e. {x} = x − bxc). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v1, . . . , vk there exists a real number t such that 1/(k + 1) ≤ {vit} ≤ k/(k + 1) for i = 1, . . . , k.

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A short proof for 6 lonely runners

If x is a real number, we denote by < x >∈ [0, 1) the fractional part of x: < x >= x− E(x), where E(x) is the integer part of x. We give a simple proof of the following version of the Lonely Runner Conjecture: if v1,...,v5 are positive integers, there exists a real number t such that < tvi >∈ [1/6, 5/6] for each i in {1, ..., 5}. Our proof requires a careful study of the different congruence cl...

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A Note on Lacunary Lonely Runners

We give a simple argument proving the lonely runner conjecture in the case where the speed of the runners forms a certain lacunary sequence. This improves an earlier result by Pandey, and is then used to derive that for each number of runners the lonely runner conjecture is true for a set of nonzero measure in a natural probability space.

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The Lonely Runner with Seven Runners

Suppose k + 1 runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least 1/(k + 1) along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners (k ≤ 5). A formulation of the problem is...

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View-obstruction: a shorter proof for 6 lonely runners

If x is a real number, we denote by 〈x〉 ∈ [0, 1) the fractional part of x: 〈x〉 = x − E(x), where E(x) is the integer part of x. We give a simple proof of the following version of the Lonely Runner Conjecture: if v1, . . . , v5 are positive integers, there exists a real number t such that 〈tvi〉 ∈ [ 6 , 6 ] for each i in {1, . . . , 5}. Our proof requires a careful study of the different congruen...

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ژورنال

عنوان ژورنال: The Electronic Journal of Combinatorics

سال: 2001

ISSN: 1077-8926

DOI: 10.37236/1602