Name
Abstract | ||
|---|---|---|
Recently, Grynkiewicz et al. (2013), using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence a1x1+⋯+akxk≡b(modn), where a1,…,ak,b,n
(n≥1) are arbitrary integers, has a solution 〈x1,…,xk〉∈Znk with all xi 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 Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann (1839), proved an explicit formula for the number of such solutions when b=0, n=p a prime, and ∑i=1kai≡0(modp) but ∑i∈Iai⁄≡0(modp) for all 0̸≠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. 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. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1016/j.disc.2019.06.016 | Discrete Mathematics |
Keywords | Field | DocType |
Linear congruence,Distinct coordinates,Graph enumeration | Integer,Prime (order theory),Graph,Discrete mathematics,Combinatorics,Group theory,Modulo,Chinese remainder theorem,Graph enumeration,Congruence relation,Mathematics | Journal |
Volume | Issue | ISSN |
342 | 11 | 0012-365X |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Khodakhast Bibak | 1 | 13 | 6.63 |
| Bruce M. Kapron | 2 | 308 | 26.02 |
| S. Venkatesh | 3 | 53 | 7.11 |