Abstract. We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26.

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There are circular square-free words of length $n$ on three symbols for $n\ge 18$. This proves a conjecture of R. J. Simpson.

We show that Dejean's conjecture holds for n ≥ 27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.

Let $\overline{\bf t}$ be the infinite fixed point, starting with $1$, of the morphism $\mu: 0 \rightarrow 01$, $1 \rightarrow 10$. An infinite word over $\lbrace 0, 1 \rbrace$ is said to be overlap-free if it contains no factor of the form $axaxa$, where $a \in \lbrace 0,1 \rbrace$ and $x \in \lbrace 0,1 \rbrace^*$. We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a suffix which is a suffix of $\overline{\bf t}$. In particular, the lexicographically least infinite overlap-free binary word is $001001 \overline{\bf t}$.

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Abstract. We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a byproduct of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.Mathematics Subject Classification. 68R15.

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