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Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio $r>0$ such that the value of function $f$ in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it.

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  • $\begingroup$ See math.stackexchange.com/questions/304330/… It is not difficult to adapt the answer to your case. $\endgroup$ – almagest May 16 '16 at 8:21
  • $\begingroup$ Well, except that you don't specify the length of your progression. Is it meant to be infinite? $\endgroup$ – almagest May 16 '16 at 8:23
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If progression is $a+nb$, and $f(a)=p$, then $f(a+pnb)\equiv 0\pmod p$ and absolute value is more then $p$ for large $n$.

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