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Einstein's Cat
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With the lonely runner conjecture, can the runners run along a circular track of any diameter or does the conjecture require that they run along a unit circle?
The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if n runners with distinct constant speeds run around a unit circle ##\mathbb{R}/\mathbb{Z}## starting at a common time and place, then each runner will at some time be separated by a distance of at least ##1/(n+1)## from the others. In this paper we make some remarks on this conjecture.
The Lonely Runner Conjecture is a mathematical conjecture that states that if n runners run around a circular track with different speeds, there will always be a point in time where each runner is at a different distance from the starting point.
The Lonely Runner Conjecture was first proposed by mathematician John Selfridge in 1962.
No, the Lonely Runner Conjecture has not been proven. However, it has been shown to hold true for certain cases, such as when there are only two runners or when the speeds of the runners are relatively prime.
The Lonely Runner Conjecture is important because it has applications in number theory and has sparked interest in other similar conjectures and problems.
No, the Lonely Runner Conjecture has not been disproven. However, numerous attempts have been made to disprove it, and it remains an unsolved problem in mathematics.