Heisenberg s=1/2 ring consisting of a prime number of atoms

In this work it will be shown that the dimensionality of the eigenvalue problem of a Heisenberg s = 1/2 ring with a prime number N of atoms can be reduced by a factor of N. This makes small systems such as N = 5 and 7 particularly easy to solve analytically for the case of nearest-neighbor interactions, without the use of Bethe's ansatz, as well as for the general case of couplings beyond nearest neighbors. Exact expressions are given for both the magnon dispersion relations and the eigenvectors.