Displaying similar documents to “Weighted sequences in finite cyclic groups.”

On the lonely runner conjecture

Ram Krishna Pandey (2010)

Mathematica Bohemica

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Suppose k + 1 runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least 1 / ( k + 1 ) from all the others. The conjecture has been already settled up to seven ( k ≤ 6 ) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.

On non-intersecting arithmetic progressions

Régis de la Bretèche, Kevin Ford, Joseph Vandehey (2013)

Acta Arithmetica

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We improve known bounds for the maximum number of pairwise disjoint arithmetic progressions using distinct moduli less than x. We close the gap between upper and lower bounds even further under the assumption of a conjecture from combinatorics about Δ-systems (also known as sunflowers).