The Primitive Deficiency of two Primitive Strings

Let \(\Sigma \) be an alphabet of size at least 2, and let \(\textbf{Q}(\Sigma )\) denote the set of all primitive strings over \(\Sigma \). Let p and q be two distinct primitive strings over \(\Sigma \). In 1967, Lentin and Schützenberger proved that the language \(p^+q^+:= \{p^n q^m: m, n \in \mathbb {N} \setminus \{0\}\}\) contains at most one periodic string. Moreover, if \(p^n q^m\) is periodic, then either \(n = 1\) or \(m = 1\). They also showed that if \(pq^m\) is periodic, then

$$\begin{aligned} m \le \dfrac{2|p|}{|q|} + 3. \end{aligned}$$

The aim of this paper is to provide a complete characterization of all pairs of distinct primitive strings p and q such that \(pq^m\) is periodic. As a consequence, we show that if \(|p| >|q|\) and \(pq^m\) is periodic, and if t is the quotient of the integer division of|p| by|q|, then

$$\begin{aligned} m \le t + 2. \end{aligned}$$

Furthermore, if t and i are integers such that \(t \ge 2\) and \(1 \le i \le t + 2\), we show that there exist two primitive strings p and q with \(|p| >|q|\) such that t is the quotient of the integer division of|p| by|q|, and \(pq^i\) is periodic.