Dejean's conjecture holds for n>=27

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. Repetitions in words have been studied since the beginning of the previous century [14, 15]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 6, 7, 8, 10]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx such that x is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifying the repetitive threshold for each alphabet size n > 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The author is supported by an NSERC Discovery Grant. The author is supported by an NSERC Postdoctoral Fellowship.