  Signatures of Majorana fermion bound states in one-dimensional topological
insulator (TI) nanowires with proximity effect induced superconductivity are
studied. The phase diagram and energy spectra are calculated for single TI
nanowires and it is shown that the nanowires can be in the topological
invariant phases of winding numbers $W=0, \pm 1$, and $\pm 2$ corresponding to
the cases with zero, one and two pairs of Majorana fermions in the single TI
nanowires. It is also shown that the topological winding numbers, i.e., the
numbers of pairs of Majorana fermions in the TI nanowires can be extracted from
the transport measurements of a Josephson junction device made from two TI
nanowires, while the sign in the winding numbers can be extracted using a
superconducting quantum interference device (SQUID) setup.
