Abstract

Let π(w) denote the minimum period of the word w, let w be a primitive word with period π(w)<|w|, and let z be a prefix of w. It is shown that if π(wz)=π(w), then |z|<π(w)-gcd(|w|,|z|). Detailed improvements of this result are also proven. Finally, we show that each primitive word w has a conjugate w'=vu, where w=uv, such that π(w')=|w'| and |u|<π(w). As a corollary we give a short proof of the fact that if u, v, w are words such that u² is a prefix of v², and v² is a prefix of w², and v is primitive, then |w|>2|u|.

Keywords: combinatorics on words, periods

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