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[ThEdArKlOrD]

For fixed $\epsilon >0$, determine whether or not, there exist polynomial $P(x)\in \mathbb{R}[x]$ such that $|P(n)-p_n|<\epsilon$ where $p_n$ is $n^{th}$ prime numbers for all positive integer $n$

[Ego]

For fixed $\epsilon >0$, determine whether or not, there exist polynomial $P(x)\in \mathbb{R}[x]$ such that $|P(n)-p_n|<\epsilon$ where $p_n$ is $n^{th}$ prime numbers for all positive integer $n$

Sorry if i'm misunderstanding but if we choose $\epsilon$ such that $0 < \epsilon < |P(1) - 2|$ then the answer is no. I think the problem should be "... there exists $n_{0}$ such that $|P(n) - p_{n}| < \epsilon$ forall $n > n_{0}$'. It's nicer