On the number of $k$-powers in a finite word

Abstract

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound of the number of {\em $k$-powers} in a finite word for any integer $k\ge 3$. By {\em $k$-power}, we mean a word of the form $\underset{k\text{times}}{\underbrace{uu...u}}$.