An almost all result on $q_1{q}_2\equiv c(\mathrm{mod}q)$

Abstract

In this paper we consider the congruence equation $q_1{q}_2\equiv c(\mathrm{mod}q)$ with $a<q_1\le a+q^{1/2+\epsilon }$ and $b<q_2\le b+q^{1/2+\epsilon }$ and show that it has solution for almost all $a$ and $b$. Then we apply it to a question of Fujii and Kitaoka as well as generalize it to more variables. At the end, we will present a new way to attack the above congruence equation question through higher moments.

12 pages, the author welcomes any ideas on the higher moment attack