On a restricted linear congruence

Abstract

Let $b,n\in \mathbb{Z}$, $n\ge 1$, and ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_{\tau (n)}$ be all positive divisors of $n$. For $1\le l\le \tau (n)$, define ${\mathcal{C}}_l:=\{1⩽x⩽n:(x,n)={\mathcal{D}}_l\}$. 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 $x_1+\cdots +x_k\equiv b(\mathrm{mod}n)$, with ${\kappa }_l=|\{x_1,\dots ,x_k\}\cap {\mathcal{C}}_l|$, $1\le l\le \tau (n)$, is \begin{align*} \frac{1}{n}\mathlarger{\sum}_{d\, \mid \, n}c_{d}(b)\mathlarger{\prod}_{l=1}^{τ(n)}\left(c_{\frac{n}{{\cal D}_l}}(d)\right)^{κ_{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].

References in corpus (3)

• A multivariate arithmetic function of combinatorial and topological significance
• Adding generators in cyclic groups
• Restricted linear congruences

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• Restricted linear congruences
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• On the number of solutions of a restricted linear congruence
• On solving a restricted linear congruence using generalized Ramanujan sums
• A generalization of Schönemann's theorem via a graph theoretic method