A generalization of Sch\"{o}nemann's theorem via a graph theoretic method

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359–398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1{x}_1+\cdots +a_k{x}_k\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}n)$, where $a_1,\dots ,a_k,b,n$ ($n\geq 1$)arearbitraryintegers,hasasolution$\langle x_1,\ldots,x_k \rangle \in \Z_{n}^k$withall$x_i$ distinct. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Sch"{o}nemann almost two centuries ago(!) but his result seems to have been forgotten. Sch"{o}nemann [{\it J. Reine Angew. Math.} {\bf 1839} (1839), 231–243] proved an explicit formula for the number of such solutions when $b=0$, $n=p$ a prime, and ${\sum }_{i=1}^k{a}_i\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}p)$ but $\sum_{i \in I} a_i \not\equiv 0 \pmod{p}$forall$\emptyset \not= I\varsubsetneq \lbrace 1, \ldots, k\rbrace$. In this paper, we generalize Sch"{o}nemann’s theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems.