How many isotopes are allowed by the laws of physics?

Answer: about 7,000

An Oak Ridge National Laboratory (ORNL) and University of Tennessee team used the Department of Energy's Jaguar supercomputer to calculate the number of isotopes allowed by the laws of physics: about 7,000 possible combinations of protons and neutrons are allowed in bound nuclei with up to 120 protons, according to ORNL. Most of these nuclei have not been observed experimentally.

At left is a representation from ORNL of the nuclear landscape, with isotopes arranged by an increasing number of protons (up) and neutrons (right). The dark blue blocks represent stable isotopes. The lighter blue blocks are unstable isotopes that have been observed. The gray blocks are bound isotopes that have not been observed. Nuclear existence ends at the drip lines (orange clouds), where there is no longer enough binding energy to prevent the last nucleons from dripping off (floating blocks). Image by Andy Sproles, Oak Ridge National Laboratory.