On the equation $ \boldsymbol{n_1n_2=n_3n_4}$ restricted to factor closed sets

TL;DR

This paper investigates the number of solutions to the equation n_1 n_2 = n_3 n_4 within a bounded range, focusing on cases where the set of factors is factor-closed, especially when it consists of square-free numbers formed from distinct primes.

Contribution

It provides new bounds and insights into the solutions of the equation restricted to factor-closed sets, particularly for square-free numbers generated by distinct primes.

Findings

Derived bounds for the number of solutions N(B,F)

Analyzed the structure of solutions when F is square-free

Extended understanding of factorizations within factor-closed sets

Abstract

We study the number of solutions $N(B,F)$ of the diophantine equation $n_1n_2=n_3n_4$, 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 $F= \big\{m=p_1^{\e_1}\ldots p_k^{\e_k}, \e_j\in \{0,1\}, 1\le j\le k\big\}$, $p_1,\ldots,p_k$ being distinct prime numbers.