Theories in this many dimensions that have a factorizable n-body matrix satisfy the Yang-Baxter theorem, and thus are integrable. The Jaynes-Cummings-Hubbard model has tunneling between cavities containing atoms with this many levels. Bertrand's theorem states that a problem named for this many objects has only two potentials which lead to stable, closed (*) orbits. The Laplace-Runge-Lenz vector is used to characterize that system with this many 
Theories in this many dimensions that have a factorizable n-body matrix satisfy the Yang-Baxter theorem, and thus are integrable. The Jaynes-Cummings-Hubbard model has tunneling between cavities containing atoms with this many levels. Bertrand's theorem states that a problem named for this many objects has only two potentials which lead to stable, closed (*) orbits. The Laplace-Runge-Lenz vector is used to characterize that system with this many bodies, which is the subject of the Kepler problem. Transitions to this energy level of the hydrogen atom typically generates lines in the visible portion of the spectrum and are named for Balmer. For 10 points, give this number, which is also the number of protons in the helium atom.
Theories in this many dimensions that have a factorizable n-body matrix satisfy the Yang-Baxter theorem, and thus are integrable.
two