We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n− 3)/6⌋. Let Am = {1, 2, . . . , m}. Let ker ψ = {v ∈ A ∗ m|∀a ∈ Am, 4 divides |v|a}. (In fact, this is not Carpi’s definition of ker ψ, but rather the assertion of his Lemma 9.1.) A word v ∈ A+m is a ψ-kernel repetition if it has per...