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I was given a question in Automata that I need to prove or disprove, and I thought about this language:

$$L = \{w\in \{0, 1\}^*\mid 1\le |w| \le 2^6\}$$ Can you please help me to figure out if its minimal DFA contains only 6 states?

How can I get intuition about DFA sizes?

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    $\begingroup$ The pumping constant in the pumping lemma is the number of states in the minimal DFA. The number of states is also the number of Myhill-Nerode equivalence classes. $\endgroup$
    – Yuval Filmus
    Apr 29 at 21:51

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Suppose the minimal DFA $(Q, \delta, q_0, F)$ contains strictly less than $2^6$ states. Then there exists a state $q$ and two words $u$, $v$ such that $|u| < |v| \leqslant 2^6$ and $\delta(q_0, u) = \delta(q_0, v) = q$.

Now, can you find a contradiction?

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