what problem accounts for the nonexistence of lead-164?

Question

although lead-164 has two magic numbers, 82 protons and 82 neutrons, it is unknown.lead-208, however is known and stable.

Answer 1

What you're forgetting is exactly how much protons hate each other. Electrostatic repulsion may not exactly be on the same order of magnitude as a force as the strong force, but it certainly can and does destabilize nucleii, or we'd see isotopes of lead that are all protons and no neutrons! As you progress up the chart you find, in fact, that nucleii ideally contain about three neutrons for every two protons. This is no accident!

Keep in mind that the 'magic numbers' just refer to binding energies that are reduced, not non-existant. Just as with electron shells, having 82 neutrons is no more stable for lead with 82 protons than having, say, only 54 electrons in all its shells would be. Sure, XENON is really stable with that amount of electrons, but this ain't xenon!