The Primitive Deficiency of two Primitive Strings

Let be an alphabet of size at least 2, and let denote the set of all primitive strings over. Let p and q be two distinct primitive strings over. In 1967, Lentin and Schützenberger proved that the language contains at most one periodic string. Moreover, if is periodic, then either or. They also showed that if is periodic, then (Formula presented.) The aim of this paper is to provide a complete characterization of all pairs of distinct primitive strings p and q such that is periodic. As a consequence, we show that if and is periodic, and if t is the quotient of the integer division of|p| by|q|, then (Formula presented.) Furthermore, if t and i are integers such that and, we show that there exist two primitive strings p and q with such that t is the quotient of the integer division of|p| by|q|, and is periodic.