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 $$n\in\mathbb{Z}$$.
That is false, take $$n=k\times a_0, k\in\mathbb Z$$, then $$a_0|f(n)$$.
• Which would be fine if $a_0 = \pm 1$ (which cannot be because $|a_0|=f(0)$ is supposed to be a prime). – Sil Oct 13 '19 at 6:50
• But then even $f(0)$ would not be prime ! – Martund Oct 13 '19 at 8:54