ABSTRACT
Another restriction of potential interest is imposing the condition that all xi are distinct. We call such congruences alldiff congruences. Quite surprisingly, the number of solutions of this type of congruences was first considered, in a special case, by Schönemann in 1839(!) but his result seems to have been overlooked. In Chapter 6, we generalize Schönemann's theorem via a graph theoretic method which may be also of independent interest. We also consider unweighted alldiff congruences and, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, give an explicit formula for the number of their solutions. We even go further and discuss applications/connections to several combinatorial problems.
