  Recently, ultra-small-diameter Single Wall Nano Tubes with diameter of $ \sim
0.4 nm$ have been produced and many unusual properties were observed, such as
superconductivity, leading to a transition temperature $T_c\sim 15^oK$, much
larger than that observed in the bundles of larger diameter tubes.
  By a comparison between two different approaches, we discuss the issue
whether a superconducting behavior in these carbon nanotubes can arise by a
purely electronic mechanism. The first approach is based on the Luttinger Model
while the second one, which emphasizes the role of the lattice and short range
interaction, is developed starting from the Hubbard Hamiltonian. By using the
latter model we predict a transition temperature of the same order of magnitude
as the measured one.
