Packing dimers on (2p+1) x (2q+1) lattices

We use computational method to investigate the number of ways to pack dimers on m x n lattices, where both m and n are odd. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on (2k+1) x (2k+1) odd square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on 2k x 2k even square lattices, for which an exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd-by-odd lattice strips with any width n > or =1. This logarithmic term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer-dimer problem might be solvable.