We report the results of studies of the magnetoresistance (MR) and electric field effect (EFE) of single-crystal Bi nanowires with diameter 40 nm<d<800 nm at low temperatures. Single-crystal nanowire samples were prepared by the Taylor-Ulitovsky technique; they were cylindrical single crystals with the orientation along the wire axis where the C3 axis was inclined at an angle of 70o to the wire axis. According to theory of S. Murakami, bismuth bilayers can exhibit the quantum spin Hall effect. A Bi crystal can be viewed as a stacking of bilayers with a honeycomblike lattice structure along the [111] direction. Using electric field effect in measurements of quantum magnetic oscillations we have confirmed the dependence of the surface states on the diameter of the Bi nanowires.