On the equation ${\mathbf{n}}_1{\mathbf{n}}_2={\mathbf{n}}_3{\mathbf{n}}_4$ restricted to factor closed sets

Abstract

We study the number of solutions $N(B,F)$ of the diophantine equation $n_1{n}_2=n_3{n}_4$, where $1\le n_1\le B$, $1\le n_3\le B$, $n_2,n_4\in F$ and $F\subset [1,B]$ is a factor closed set. We study more particularly the case when $F= \big\{m=p_1^{\e_1}\ldots p_k^{\e_k}, \e_j\in \{0,1\}, 1\le j\le k\big\}$, $p_1,\dots ,p_k$ being distinct prime numbers.

This revised and corrected version has been published in Acta Math. Sinica , 2018 , vol. 34 No. 10, pp. 1517--1530