  Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and
[F:Q] is odd. For delta \in F^times, let [delta] denote its class in
F^times/F^{times p}. In this paper, we show
  Main Theorem. There are infinitely many classes [delta]\in F^times/F^{times
p} such that the twisted affine Fermat curves W_delta: X^p+Y^p=delta have no
F-rational points.
