  We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum
spin chain with periodic boundary conditions. We formulate a conjecture for the
number of solutions with pairwise distinct roots of these equations, in terms
of numbers of so-called singular (or exceptional) solutions. Using homotopy
continuation methods, we find all such solutions of the Bethe equations for
chains of length up to 14. The numbers of these solutions are in perfect
agreement with the conjecture. We also discuss an indirect method of finding
solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly
comment on implications for thermodynamical computations based on the string
hypothesis.
