On the number of solutions of a restricted linear congruence (1708.04939v1)

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]