Non-primitive words of the form pq^m

¹ King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics PO Box 5046, Dhahran 31261, Saudi Arabia.
² University Tunis-El Manar. Faculty of Sciences of Tunis, Department of Mathematics, 2092 Tunis, Tunisia.
othechi@yahoo.com

Received: 8 November 2017
Accepted: 14 November 2017

Abstract

Let p,q be two distinct primitive words. According to Lentin−Schützenberger [9], the language p⁺q⁺ contains at most one non-primitive word and if pq^m is not primitive, then $\mathit{m}\mathrm{\le }\frac{\mathrm{2}\mathrm{\left|}\mathit{p}\mathrm{\right|}}{\mathrm{\left|}\mathit{q}\mathrm{\right|}}\mathrm{+}\mathrm{3}$. In this paper we give a sharper upper bound, namely, $\mathit{m}\mathrm{\le }\mathrm{\lfloor }\frac{\mathrm{\left|}\mathit{p}\mathrm{\right|}\mathrm{-}\mathrm{2}}{\mathrm{\left|}\mathit{q}\mathrm{\right|}}\mathrm{+}\mathrm{2}\mathrm{\rfloor }\mathit{,}$ where ⌊ x ⌋ stands for the floor of x.