On the number of squares in a finite word

A {\em square} is a word of the form $uu$. In this paper we prove that for a given finite word $w$, the number of distinct square factors of $w$ is bounded by $\mathrm{\mid }w\mathrm{\mid }-\mathrm{\mid }\text{Alphabet}(w)\mathrm{\mid }+1$, where $\mathrm{\mid }w\mathrm{\mid }$ denotes the length of $w$ and $\mathrm{\mid }\text{Alphabet}(w)\mathrm{\mid }$ denotes the number of distinct letters in $w$. This result answers a conjecture of Fraenkel and Simpson stated in 1998.