The above second-quantized model takes the particles to be bosons. If the particles are taken to be fermions, then the Pauli exclusion principle prohibits multi-particle states which include squares of primes. By the spin–statistics theorem, field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator (−1) has a very concrete realization in this model as the Möbius function μ μ ( n ) , in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.