A possible strategy for the long-term sustainability of nuclear energy is to use fast reactors with closed fuel cycles, i.e. to recover all the actinides in the spent fuel (including the MA, Minor Actinides) and recycle them in the reactor itself. In this way the fuel cycle can have as input only depleted (or natural) Uranium and as final waste only the fission products and the losses in the fuel reprocessing (e.g. 0.1%). The feasibility of this approach depends on the equilibrium fuel configuration (the fuel composition attained after infinite cycles), when the final isotopic composition of the fuel after a certain irradiation time t1 and cooling time t2 turns out to be exactly the same as the one which has been loaded (except for Uranium). Two independent mathematical methods are here outlined for the solution of the fuel equilibrium composition. These methods are then validated by the codes FISPACT and MCNPX and applied to the European lead fast reactor ELSY. The fraction (on the total actinides) at equilibrium turns out to be in this case 0.9% for the MA and 17.2% for the Pu. The keff and the βeff are then calculated showing the promising features of this approach.