On the number of solutions of a restricted linear congruence

Abstract

Consider the linear congruence equation $$a_1^s{x}_1+\dots +a_k^s{x}_k\equiv b(\text{mod }{n}^s)\text{ where }{a}_i,b\in \mathbb{Z},s\in \mathbb{N}$$ 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 $⟨x_1,\dots ,x_k⟩\in {\mathbb{Z}}^k$ 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]

References in corpus (5)

• Adding generators in cyclic groups
• Restricted linear congruences
• On a restricted linear congruence
• On solving a restricted linear congruence using generalized Ramanujan sums
• The Discrete Fourier Transform of $(\mathbf{r},\mathbf{s})-$even functions