On Sets of Square-Free Numbers

Let N(x)=N (x,d₁,d₂,.....,d_r-1) denote the number of groups of square-free numbers q₁,q₂,...,q_r not exceeding x, such that q_m-q₁=d_m-1. (m=2, 3, . . ., r). Further let f(p) be the number of different residue classes modulus p² and not congruent to 0 contained in the set d₁,d₂,.. d_r-1.