PILLAI’S PROBLEM ON CONSECUTIVE INTEGERS

Abstract. For integers m ≥ 2 and d ≥ 1, we study the set Sm of m consecutive integers which satisfies the property that for each x ∈ Sm there exists y ∈ Sm such that gcd(x, y)> d. This problem was first posed and studied by S. S. Pillai for the case d = 1. In this article, we elaborate on an argument of T. Vijayaraghavan for d = 1 and of Y. Caro for d ≥ 1. For an integer m ≥ 2, let Sm be a set of m consecutive integers. Let d ≥ 1 be an integer. We say that the set Sm has property Pd if there exists an element x ∈ Sm such that gcd(x, y) ≤ d for all y ∈ Sm with y 6 = x. In this case, we also say that the element x has property Pd. When n