On disjoint residue classes

Abstract

The purpose of this note is to show that, if n 1,…, n k are positive integers, and for each d ϵ Z + satisfying f( d) ⩽ k− 2 or a weaker condition d ⩽ 2 k−2 (where f( d) = ∑ r i = 1 α i ( P i − 1) if Π r i = 1 P α i i is the prime factorization of d), the number of pairs { i,j} (1 ⩽ i < j ⩽ k) with gcd ( n i , n j = d is less than (d + 7) 8 , then there exist integers a 1,…, a k such that the residue classes a 1(mod n 1),…, a k are pairwise disjoint. We conjecture that (d + 7) 8 can be replaced by 2 d−1.