  We investigate the existence of time-periodic solutions of the Dirac equation
in the Kerr-Newman background metric. To this end, the solutions are expanded
in a Fourier series with respect to the time variable $t$ and the Chandrasekhar
separation ansatz is applied so that the question of existence of a
time-periodic solution is reduced to the solvability of a certain coupled
system of ordinary differential equations. First, we prove the already known
result that there are no time-periodic solutions in the non-extreme case. Then
it is shown that in the extreme case for fixed black hole data there is a
sequence of particle masses $(m_N)_{N\in\mathbb N}$ for which a time-periodic
solution of the Dirac equation does exist. The period of the solution depends
only on the data of the black hole described by the Kerr-Newman metric.
