Tight Instances of the Lonely Runner

The Lonely Runner Conjecture of J. Wills asserts the following. For any vector v ∈ (R−{0})n−1 there exists t ∈ R such that every component of tv has distance at least 1/n to the nearest integer. We study those vectors v for which the bound of this conjecture is attained. In particular, we construct an infinite family of such tight vectors. This family, plus three sporadic examples, constitute all known tight vectors. We completely characterize a subfamily: those tight vectors obtained from v = 〈1, 2, . . . , n − 1〉 by scaling one entry by a positive integer. The characterization motivates the problem of finding the least positive integer which has a common factor with every integer in the interval [a, b]. We solve this problem when b ≥ 2a.