# Problem 72: Three Square Prefixes

The combinatorial analysis of square prefixes of a word leads to several consequences useful to design algorithms related to periodicities.

Three non-empty words $u$, $v$ and $w$ satisfy the square-prefix condition if $u^2$ is a proper prefix of $v^2$ and $v^2$ a proper prefix of $w^2$.

Show that if $u^2$, $v^2$ and $w^2$ satisfy the square-prefix condition and $|w| \leq 2|u|$ then $u,v,w \in z^2z^*$ for some word $z$.

The conclusion implies in particular that $u$ is not primitive. In fact, this implication holds true if both the square-prefix condition and the inequality $|w|\lt |u|+|v|$ are met (Three-Square-Prefix lemma). But the statement in the above question has a stronger conclusion that says we are essentially in the trivial situation where $w^2=\sa{a}^k$ or the like.

Give infinitely many examples of word triples that satisfy the square-prefix condition and for which both $|u|+|v|=|w|$ and $u$ is primitive.

The next question provides a consequence of the Three-Square-Prefix lemma or of the first statement. The exact upper bound or even a tight bound on the concerned quantity is still unknown.

Show that less than $2|x|$ (distinct) primitively rooted squares can be factors of a word $x$.

Another direct consequence of the Three-Square-Prefix lemma is that a word of length $n$ has no more than $\log_\Phi n$ prefixes that are primitively rooted squares. The golden mean $\Phi$ comes from the recurrence relation for Fibonacci numbers in the second question.

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