carbon nanotubes. The energy band structure of a carbon nanotube may ordinarily be obtained by “zone-folding” the nearest neighbour tight-binding approximation of a flat graphite sheet. Due to the large curvature of a small-radius tube, however, this procedure fails to accurately predict the metallic character of any zigzag nanotube whose diameter is less than 5˚A. To this end, a third-nearest neighbour tight-binding approximation sensitive to the curvature of the tube is presented and an analytic expression for the energy bands is derived valid for all carbon nanotubes. Electron-electron interactions in small-radius zigzag carbon nanotubes are then considered by including a weak on-site Hubbard term in the tight-binding Hamilton operator. At half-filling, the system is mapped onto a set of three one-dimensional chains with alternating hopping coefficients and the low-energy physics are studied. A controlled renormalization-group analysis shows that the system flows to a strong-coupling fixed point where 1 charge mode and 2 spin modes are gapped. This fixed point also exists in the two-chain Hubbard ladder and superconducting fluctuations have previously been found to be dominant. These results support previous experimental evidence of intrinsic superconductivity in a 4˚A-diameter nanotube and suggest that both the (5, 0) and the larger (6, 0) nanotubes should be superconducting.