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The electronic, mechanical, and optical properties of (3,3), (5,5), (7,7) and (9,9) armchair and (4,0), (6,0), (8,0), (10,0) and (12,0) zigzag boron phosphide nanotubes (BPNT's) have been investigated using density functional theory calculations. We find that, with the exception of the (5,5) nanotube, they are direct band gap semiconductors, as the BP monolayer. The (3,3) and (4,0) nanotubes are verified to be unstable through phonon band structure calculations. The 1D stretching moduli and elastic moduli show that the BP NT's are as stiff as carbon nanotubes of comparable diameters. The absorption spectra of the BP nanotubes have thresholds in the visible region, differently from carbon and boron nitride nanotubes of comparable dimensions.