On primitive words with non-primitive product

Let be an alphabet of size n ≥ 2. Our goal in this paper is to give a complete description of primitive words p≠q over such that pq is non-primitive. As an application, we will count the cardinality of the set ℰ(l,) of all couples (p, q) of distinct primitive words such that |p| = |q| = l and pq is non-primitive, where l is a positive integer. Then we give a combinatorial formula for the cardinality ε(n, l) of this set. The density in {(p, q) : p, q are distinct primitive words and |p| = |q| = l} of the set ℰ(l,) is also discussed