magic numbers
Some numbers of protons or neutrons make a nucleus conspicuously special: unusually tightly bound, unusually abundant, reluctant to capture another nucleon, with a large gap up to the first excited state. Just as atoms with 2, 10, 18 electrons (the noble gases) are chemically inert because they have closed electron shells, nuclei with certain nucleon counts have closed nuclear shells. Those counts are the magic numbers.
The established magic numbers are 2, 8, 20, 28, 50, 82, and 126. They emerge directly from the nuclear shell model: they are exactly the totals at which the single-particle energy levels, including the crucial spin-orbit splitting, group into well-separated bunches, so that filling a bunch leaves a large energy gap before the next level. A nucleus is doubly magic when both its proton number and its neutron number are magic, giving maximum stability. The magic numbers are why the naive smooth semi-empirical mass formula, which knows nothing of shells, systematically underbinds these nuclei.
You meet magic numbers everywhere in nuclear science. The doubly magic nuclei helium-4, oxygen-16, calcium-40 and -48, and lead-208 are landmarks of stability; tin (Z = 50) has the most stable isotopes of any element; and the anticipated 'island of stability' for superheavy elements is a hunt for the next proton magic number beyond 82 (often predicted near 114 or 120). Ongoing research shows magic numbers can shift or dissolve far from stability, so they are robust but not immutable.
Lead-208 is doubly magic: 82 protons and 126 neutrons. It is the heaviest stable nucleus, and it caps the natural decay chains of thorium, uranium and actinium.
Every natural heavy decay chain ends at a doubly magic lead isotope.
The magic numbers are not the same as filled harmonic-oscillator shells; getting 28, 50, 82, 126 right requires the strong spin-orbit interaction. They are also not perfectly fixed and can change for very neutron-rich nuclei.