Remarks on a paper of J. Barát and P.P. Varjú

Abstract

Let {d i n+b i :n∈ℤ} i∈I be a family of disjoint arithmetic progressions covering the integers. Barát and Varjú [1] have proved that if d i =p 1 α 1 p 2 α 2 for two prime numbers p 1 , p 2 and integers des α 1 ,α 2 ≥0, then there exist j≠i such that d i |d j . We show that this result remains true if d i =p 1 α 1 ⋯p n α n for a fixed set {p 1 ,⋯,p n } of n prime numbers.