The Lowest Order Constrained Variational (LOCV) method is one of the approximated approaches for the calculation of the nuclear many-body energy. In this method, the energy of the nucleonic system is calculated by truncating the cluster expansion of the energy at the two-body level, in the presence of the normalization constraint. In 2001, a fixed number of nucleons was studied by the Periodic Box Fermi Hypernetted Chain (PBFHNC) technique. In 2013, similar to the PBFHNC calculations, the LOCV approach was reformulated for a fixed number of nucleons in a periodic box (PBLOCV). Fortunately, the approximated PBLOCV results lied near those of PBFHNC. However, the energy in the PBLOCV approach was calculated in the two-body cluster approximation. In the present research, the energy of a fixed number of nucleons in a period box is obtained for the Bethe homework and the otentials, using the LOCV correlation functions. The ratio of the three-body cluster energy to that of two-body ranges from 0.13 to 1.9. This means that the contribution of the three-body cluster energy is relatively smaller than that of two-body. o, it is shown that the PBLOCV approximation is valid at the lowest order. The key point of the mentioned achievement is the special behavior of the LOCV correlation functions, which are the inputs of the PBLOCV computations. The LOCV correlation functions satisfy the normalization constraint of the radial distribution function. As a result, the many-body cluster terms can be ignored in the cluster expansion of the energy.