On the number of solutions of a restricted linear congruence

TL;DR

This paper generalizes the counting of solutions for a restricted linear congruence to arbitrary powers, using generalized Ramanujan sums, extending previous results that only covered the case when s=1.

Contribution

It derives a new formula for the number of solutions of a restricted linear congruence for any natural number s, involving generalized Ramanujan sums.

Findings

Provides a formula for solutions count with arbitrary s

Extends previous s=1 results to general s

Uses generalized Ramanujan sums in the solution formula

Abstract

Consider the linear congruence equation $${a_1^{s}x_1+\ldots+a_k^{s} x_k \equiv b\,(\text{mod } n^s)}\text { where } a_i,b\in\mathbb{Z},s\in\mathbb{N}$$ Denote by $(a,b)_s$ the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. Given $t_i|n$, we seek solutions $\langle x_1,\ldots,x_k\rangle\in\mathbb{Z}^k$ for this linear congruence with the restrictions $(x_i,n^s)_s=t_i^s$. Bibak et al. [J. Number Theory, 171:128-144, 2017] considered the above linear congruence with $s=1$ and gave a formula for the number of solutions in terms of the Ramanujan sums. In this paper, we derive a formula for the number of solutions of the above congruence for arbitrary $s\in\mathbb{N}$ which involves the generalized Ramanujan sums defined by E. Cohen [Duke Math. J, 16(85-90):2, 1949]