What are the 'magic numbers' in nuclear physics, and why are they so powerful?

Nuclear "magic numbers" (2, 8, 20, 28, 50, 82, 126) represent quantities of protons or neutrons that create unusually stable atomic nuclei. These numbers correspond to filled nuclear shells, similar to electron shells in atoms, where the strong force binding particles together is higher than expected, resulting in spherical nuclei resistant to radioactive decay. Lead-208, with 82 protons and 126 neutrons, exemplifies this stability as the heaviest stable element known.