A generalization of Schönemann's theorem via a graph theoretic method
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Recently, Grynkiewicz et al. [ Israel J. Math. 193 (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence a_1x_1+... +a_kx_k≡ b n, where a_1,...,a_k,b,n (n≥ 1) are arbitrary integers, has a solution 〈 x_1,...,x_k 〉∈_n^k with all x_i distinct modulo n. 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 Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann [ J. Reine Angew. Math. 1839 (1839), 231--243] proved an explicit formula for the number of such solutions when b=0, n=p a prime, and ∑_i=1^k a_i ≡ 0 p but ∑_i ∈ I a_i ≡ 0 p for all I 1, ..., k. In this paper, we generalize Schönemann'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 recently obtained by Ardila et al. [ Int. Math. Res. Not. 2015 (2015), 3830--3877]. 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.
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