  We consider Josephson junctions on surfaces of three dimensional topological
insulator nanowires. We find that in the presence of a parallel magnetic field,
short junctions on nanowires show signatures of a perfectly transmitted mode
capable of supporting Majorana fermions. Such signatures appear in the
current-phase relation in the presence or absence of the fermion parity
anomaly, and are most striking when considering the critical current as a
function of flux \Phi, which exhibits a peak at \Phi=h/2e. The peak sharpens in
the presence of disorder at low but finite chemical potentials, and can be
easily disentangled from weak-antilocalization effects. The peak also survives
at small but finite temperatures, and represents a realistic and robust
hallmark for perfect transmission and the emergence of Majorana physics inside
the wire.
