The range of non-linear natural polynomials cannot be context-free
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Suppose that some polynomial f with rational coefficients takes only natural values at natural numbers, i.e., L={f(n)| n∈ N}⊂ N. We show that the base-k representation of L is a context-free language if and only if f is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.
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