A rational function quantum equivalence relation on the set of all prime numbers

In this paper, we prove new results which make it possible to vastly simplify solution of the problem of Nathanson concerning the enlargement of support bases and thus of the corresponding semigroup supports of rational function solutions to the quantum functional equations arising from multiplication of quantum integers. In particular, we establish a certain quantum equivalence relation on the set of all prime numbers which encodes the symmetry of these solutions and thus solve a problem which simplify in crucial ways the criteria for the problem posed by Nathanson concerning extensions of the support bases of these solutions.