Dimer adsorption on an infinitely long (100) or square lattice nanotube of fixed lattice constant and arbitrary number M of sites in its normal cross section is considered with first- and second-neighbor interaction energies, V and W, respectively. An increase in M keeping the lattice constant fixed corresponds to a nanotube with increasing diameter. The system is at thermodynamic equilibrium and relatively low temperature in order to determine all of the possible crystallization patterns or phases that may exist. Under these conditions, the energy phase space diagram is two-dimensional, and the dimensionless parameters are u = W/vertical bar V vertical bar and v = mu/vertical bar V vertical bar, where mu is the sum of the chemical potential energy, mu', of a dimer and the dimer-substrate interaction energy V-0. The low temperature energy phase diagram is numerically generated using a transfer matrix method adapted to the present problem. For both attractive and repulsive first-neighbors, it is M-independent for M even, and its M-dependence when M is odd is established. As a consistency check, the infinite odd-M limit matches exactly the M-independent phase diagram for even M. The knowledge of the boundaries of the regions in the phase diagram where these phases occur and the knowledge of their respective occupational configurations are particularly useful information when experimental data on dimer adsorption on square nanotubes become available.