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I have recently started to learn a bit of number theory, and I just wanted to pose the following problem out of curiosity. I don't know what the answer is myself. Here is the problem.

Prove or disprove: there exists a function $f:\mathbb{Z}\rightarrow\mathbb{N}\cup \{0\}$, $f(n)= |\sum_{i=0}^{N}a_in^i |$, where each $a_i\in\mathbb{Z}$, such that $f(n)$ is prime for all $n\in\mathbb{Z}$.

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That is false, take $n=k\times a_0, k\in\mathbb Z$, then $a_0|f(n)$.

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    $\begingroup$ Which would be fine if $a_0 = \pm 1$ (which cannot be because $|a_0|=f(0)$ is supposed to be a prime). $\endgroup$ – Sil Oct 13 '19 at 6:50
  • $\begingroup$ But then even $f(0)$ would not be prime ! $\endgroup$ – Martund Oct 13 '19 at 8:54
  • $\begingroup$ Exactly, that's my point. $\endgroup$ – Sil Oct 13 '19 at 8:56

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