An upper bound of the number of distinct powers in binary words

Let u be a finite word and r be a positive integer, a power is a word of the form uu…u︸rtimes. Particularly, a square is a word of the form uu. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by its length. This conjecture was proved recently by Brlek and Li. Besides, a stronger upper bound for binary words was conjectured by Jonoska, Manea and Seki by stating that, for a word of length n over the alphabet {a,b}, if k is the least of the number of a's and the number of b's and k≥2, then the number of distinct squares is upper bounded by [Formula presented]. In this article, we prove this conjecture by giving a stronger statement on the number of distinct powers in a binary word.