Mathematica Bohemica, Vol. 146, No. 1, pp. 47-54, 2021
A formula for the number of solutions of a restricted linear congruence
K. Vishnu Namboothiri
Received December 25, 2018. Published online January 15, 2020.
Abstract: Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\pmod{n^s}$ for $b\in\mathbb Z$, $n,s\in\mathbb N$. Let $(a,b)_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in\mathbb N$ dividing $a$ and $b$ simultaneously. Let $d_1,\ldots, d_{\tau(n)}$ be all positive divisors of $n$. For each $d_j\mid n$, define $\mathcal{C}_{j,s}(n) = \{1\leq x\leq n^s (x,n^s)_s = d^s_j\}$. K. Bibak et al. (2016) 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 $x_i$ and prove that for the above linear congruence, the number of solutions is
$\frac1{n^s}\sum\limits_{d\mid n}c_{d,s}(b)\prod\limits_{j=1}^{\tau(n)}\Bigl(c_{n/{d_j},s}\Bigl(\frac{n^s}{d^s}\Big)\Big)^{g_j}$
where $g_j = |\{x_1,\ldots, x_k\}\cap\mathcal{C}_{j,s}(n)|$ for $j=1,\ldots, \tau(n)$ and $c_{d,s}$ denotes the generalized Ramanujan sum defined by E. Cohen (1955).
Affiliations: K. Vishnu Namboothiri, Department of Mathematics, Government College, Ambalapuzha, Alappuzha 688 561, Kerala, India, and Department of Collegiate Education, Government of Kerala, 6th Floor, Vikas Bhavan, Palayam, Thiruvananthapuram 695 033, Kerala, India, e-mail: kvnamboothiri@gmail.com