A generalization of Schönemann's theorem via a graph theoretic method

Abstract Recently, Grynkiewicz et al. (2013), using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence a 1 x 1 + ⋯ + a k x k ≡ b ( mod n ) , where a 1 , … , a k , b , n ( n ≥ 1 ) are arbitrary integers, has a solution 〈 x 1 , … , x k 〉 ∈ Z n k with all 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 Schonemann almost two centuries ago(!) but his result seems to have been forgotten. Schonemann (1839), proved an explicit formula for the number of such solutions when b = 0 , n = p a prime, and ∑ i = 1 k a i ≡ 0 ( mod p ) but ∑ i ∈ I a i ⁄ ≡ 0 ( mod p ) for all 0 ≠ I ⊊ { 1 , … , k } . In this paper, we generalize Schonemann’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.

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