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Results of cold and hot asymmetric nuclear matter and pure neutron matter calculations are presented. The (BHF) approximation + two body density dependent Skyrme potential which is equivalent to three - body interaction are used. Various modern nucleon-nucleon (NN) potentials are used as follows: CD-Bonn potential, Nijm1 potential, Reid 93 potential and Argonne V18 potential are used in the framework of (BHFA). The bulk properties of asymmetric nuclear matter are computed such as the EOS at (T = 0), pressure at (T = 0, 5 and 10 MeV), single particle potential, entropy at (T = 5 and 10 MeV), free energy at (T = 5 and 10 MeV), nuclear matter incompressibility and the symmetry energy. Also the bulk properties of pure neutron matter are computed such as the EOS at (T = 0), pressure at (T = 0, 3 and 6 MeV), single particle potential, entropy at (T = 3 and 6 MeV), free energy at (T = 3 and 6 MeV). The results are compared with B. Freidman and V. R. PandhariPand [45]. By this method good agreement is obtained in comparison with previous theoretical estimates and experimental data.