The relation between F(R) gravity and Einstein-conformally invariant Maxwell source

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.