A Generalization Of Schonemann'S Theorem Via A Graph Theoretic Method

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) equivalent to b (mod n), where a(1), ..., a(k), b, n (n >= 1) are arbitrary integers, has a solution < x1, ..., x(k)> is an element of 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(l) 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 Sigma(k)(i=1) a(i) equivalent to 0 (mod p) but Sigma i is an element of l a(i) not equivalent to 0 (mod p) for all empty set not equal I not subset of {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. (C) 2019 Elsevier B.V. All rights reserved.