On a restricted linear congruence

TL;DR

This paper provides a concise proof for counting solutions to linear congruences with restrictions based on divisors, using Fourier transforms and Ramanujan sums, with broad applications in various fields.

Contribution

It introduces a new explicit formula for solutions of restricted linear congruences, generalizing several previous results with a novel proof technique.

Findings

Derived an explicit formula involving Ramanujan sums for solution counts

Unified and extended previous results in number theory and related fields

Applicable to problems in cryptography, combinatorics, and computer science

Abstract

Let $b,n\in \mathbb{Z}$, $n\geq 1$, and ${\cal D}_1, \ldots, {\cal D}_{\tau(n)}$ be all positive divisors of $n$. For $1\leq l \leq \tau(n)$, define ${\cal C}_l:=\lbrace 1 \leqslant x\leqslant n \; : \; (x,n)={\cal D}_l\rbrace$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\cdots +x_k\equiv b \pmod{n}$, with $\kappa_{l}=|\lbrace x_1, \ldots, x_k \rbrace \cap {\cal C}_l|$, $1\leq l \leq \tau(n)$, is \begin{align*} \frac{1}{n}\mathlarger{\sum}_{d\, \mid \, n}c_{d}(b)\mathlarger{\prod}_{l=1}^{\tau(n)}\left(c_{\frac{n}{{\cal D}_l}}(d)\right)^{\kappa_{l}}, \end{align*} where $c_{d}(b)$ is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory {\bf 133} (2013), 705--718], one of the main results of the paper by Sander [J. Number Theory {\bf 129} (2009), 2260--2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory {\bf 10} (2014), 1355--1363].