How Many Squares Can a String Contain?

AbstractAll our words (strings) are over afixedalphabet. A square is a subword of the formuu=u2, whereuis a nonempty word. Two squares aredistinctif they are of different shape, not just translates of each other. A worduisprimitiveifucannot be written in the formu=vjfor somej⩾2. A squareu2withuprimitive isprimitive rooted. LetM(n) denote the maximum number of distinct squares,P(n) the maximum number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduceM(n)<2nfor alln>0, andP(n)=n−o(n) for infinitely manyn