It has at times been indicated that Landau introduced neutron stars in his classic paper of 1932. This is clearly impossible because the discovery of the neutron by Chadwick was submitted more than one month after Landau's work. Therefore, and according to his calculations, what Landau really did was to study white dwarfs, and the critical mass he obtained clearly matched the value derived by Stoner and later by Chandrasekhar. The birth of the concept of a neutron star is still today unclear. Clearly, in 1934, the work of Baade and Zwicky pointed to neutron stars as originating from supernovae. Oppenheimer in 1939 is also well known to have introduced general relativity (GR) in the study of neutron stars. The aim of this note is to point out that the crucial idea for treating the neutron star has been advanced in Newtonian theory by Gamow. However, this pioneering work was plagued by mistakes. The critical mass he should have obtained was $6.9\,M_\odot$, not the one he declared, namely, $1.5\ M_\odot$. Probably, he was taken to this result by the work of Landau on white dwarfs. We revise Gamow's calculation of the critical mass regarding calculational and conceptual aspects and discuss whether it is justified to consider it the first neutron-star critical mass. We compare Gamow's approach to other early and modern approaches to the problem.