Diophantine Equation for the Hall Conductance of Interacting Electrons on a Torus

We derive a Diophantine equation for the Hall conductance of $N$ interacting electrons moving on a torus. The equation holds for general background fields, including inhomogeneous magnetic fields, and random substrates, but is effective when combined with symmetries. For example, together with translation invariance in one direction it determines the Hall conductance uniquely and constrains the degeneracy and crossings of eigenvalues.