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Perhaps I didn't understand the largest minimal pumping length condition correctly, but suppose you have largest minimal pumping length of $k=2$. Then pick a string $s = 1^n$ with large enough $n \mod 3 = 1$; $s \in L$ (because $n$ is not a multiple of $3$). But then you have a pumpable string $vw$ made only of $1$s which has length $1$ or $2$. If it has ...


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