Periods in Extensions of Words

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|.