On solving a restricted linear congruence using generalized Ramanujan sums

Abstract

Consider the linear congruence equation $x_1+\dots +x_k\equiv b(\text{mod }n)$ for $b,n\in \mathbb{Z}$. By $(a,b{)}_s$, we mean the largest $l^s\in \mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define ${\mathcal{C}}_{j,s}=\{1\le x\le n^s|(x,n^s{)}_s=d_j^s\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\sum _{d|n}{c}_{d,s}(b)\prod _{j=1}^{\tau (n)}{\left(c_{\frac{n}{d_j},s}(\frac{n^s}{d^s})\right)}^{g_j}$$ where $g_j=|\{x_1,\dots ,x_k\}\cap {\mathcal{C}}_{j,s}|$ for $j=1,\dots \tau (n)$ and $c_{d,s}$ denote the generalized ramanujan sum defined by E. Cohen.

References in corpus (1)

• On a restricted linear congruence