Pauli exclusion principle: why does it imply that $Zapprox N$ in stable nuclides with $A<40$?

Nuclei with uneven numbers of protons and neutrons are not neccesarily unstable. There are several elements that have more or less neutrons than protons, that are stable. Silicon 28, 29, and 30 are all stable. Other light nuclei with more than one stable isotopes are O, Mg and and others.

Often, however, N=Z is the most common isotope. The reason N=Z is most popular, in terms of the Pauli Exclusion Principle is because different energy levels must be filled in order. To have a different number of protons and neutrons, the neutron may have to exist in a new shell which is higher in energy.

For light nuclei, Z <= N is preferred due to the coloumb repulsion of the protons in the nucleus. The coulomb repulsion is pushing the protons apart but the strong nuclear force is pulling the nucleis together. If more protons are added while keeping the neutron number constant we get Z > N and the nucleus becomes less stable or unstable. In this model type it is assumed that the addition of neutrons increase the nuclear force without increasing the coulomb repulsion and increasing the stability of the the nuclei.

This leads one to ask why nuclei of Z=20 and N=80 do not exist. This too can be explained in terms of the Pauli exclusion Principle. We can add neutrons to increase the nuclear force but neutrons are added according to the shell model. So as neutrons are added they must go into a higher energy level. If we try to add 60 more neutrons to a calcium (z=20) nucleus we must add the additional nuclei to higher and higher energy levels. These neutrons quickly fill levelsthat are unstable with more protons.