  In this paper, we consider the special case of $F(R)$ gravity, in which
$F(R)= R^{N}$ and obtain its topological black hole solutions in higher
dimensions. We show that, the same as higher dimensional charged black hole,
these solutions may be interpreted as black hole solutions with two event
horizons, extreme black hole and naked singularity provided the parameters of
the solutions are chosen suitably. But, the presented black hole is different
from the standard higher-dimensional Reissner-Nordstrom solutions. Next, we
focus on the conformally invariant Maxwell field coupled to Einstein gravity
and discuss about its black hole solutions. Comparing these two class of
solutions, shows that there is a correspondence between the
Einstein-conformally invariant Maxwell solutions and the solutions of $F(R)$
gravity without matter field in arbitrary dimensions.
