On a strange observation in the theory of the dimer problem Original Research Article

This is a contribution to the number theory of the dimer problem. Let An=2nBn2 (Bn>0) denote the number of dimer coverings (i.e., perfect matchings) of a 2n×2n square lattice graph. Bn turns out to be an integer. Motivated by a somewhat strange observation (see below), we investigated the residue classes Bn mod 2r. In this paper, we outline a method that, for a fixed integer r, enables these residue classes to be determined. Explicitly, Bn mod 64 is calculated and from the resulting formula the following proposition is deduced:Bn≡n+1 (mod 32)if n is even,(−1)(n−1)/2n (mod 32)if n is odd.The analogous statement mod 64 is not true: B3=29 is a counterexample.