For a dilute system of non-relativistic bosons in 3 dimensions interacting through a positive, radially symmetric, potential v with scattering length a we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1+12815πρa3+o(ρa3)), thereby proving a lower bound consistent with the Lee–Huang–Yang formula for the energy density. The proof allows for potentials with large L¹-norm, in particular, the case of hard core interactions is included. Thereby, we solve a problem in mathematical physics that had been a major challenge since the 1950’s.