On f(R) theories equivalent to general relativity for a specific form of   the Ricci scalar
  We prove the necessary and the sufficient condition on the f(R) function leading to the equivalence of general relativity and the f(R) theory in the sense that for a spacetime with specific Ricci scalar, GR and the f(R) theory will necessarily have the same solutions. We show how this condition and its consequences allow to get such functions and give the possibility to fixe conditions on the parameters of any class of f(R) functions so that GR and this class of theories will have exactly the same solutions. The well-known model R+{\alpha}\b{eta}^{1-p}R^{p} for dark energy is presented as an illustration, as well as the FLRW metric where it's shown that there exists only one class of f(R) theories which has exactly the same GR scale factor. 
Amine Benachour
gr-qc