Abstract :
If image is a real number, we denote by image the fractional part of image: image, where image is the integer part of image. We give a simple proof of the following version of the Lonely Runner Conjecture: if image are positive integers, there exists a real number image such that image for each image in image. Our proof requires a careful study of the different congruence classes modulo 6 of the speeds image, and is simply based on the consideration of some time image maximizing the distance of image to image among the set of times image such that image for each image. In appendix, we also give elementary proofs, based on the same idea, for analogous versions of the conjecture with fewer integers.
