An almost all result on $q_1 q_2 \equiv c \pmod q$

TL;DR

This paper investigates the solvability of the congruence q_1 q_2 ≡ c mod q within specific intervals, demonstrating solutions for almost all parameters, and extends the results to multiple variables with new analytical methods.

Contribution

It provides new results on the solvability of a specific congruence in short intervals and introduces a novel approach using higher moments for analysis.

Findings

Solutions exist for almost all parameter choices within specified intervals.

Generalization of results to multiple variables.

Introduction of a higher moments method for analyzing the congruence.

Abstract

In this paper we consider the congruence equation $q_1 q_2 \equiv c \pmod q$ with $a < q_1 \leq a + q^{1/2+\epsilon}$ and $b < q_2 \leq b + q^{1/2+\epsilon}$ 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.