On the number of squares in a finite word

Let u be a nonempty finite word, a 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 |w| − | Alph(w)|, where |w| denotes the length of w and | Alph(w)| denotes the number of distinct letters in w. This result answers positively a conjecture stated by Fraenkel and Simpson in 1998 and the d-step conjecture stated by Deza, Franek and Jiang in 2011.