Diophantine-ness Preserving Functional equation(Self-made)

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Fm Jakaria
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Diophantine-ness Preserving Functional equation(Self-made)

Unread post by Fm Jakaria » Thu Apr 02, 2015 8:50 pm

Suppose that a positive integer $n$ is given. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
for any polynomial $P$ with positive integer coefficients, with at least $n$ nonzero coefficients; and for any pair of positive integers $a,b$;
if $P(a)$ divides $P(b)$; then $Q(a)$ also divides $Q(b)$, where $Q$ is the polynomial obtained by substituting coefficients $c$ of $P$ by $f(c)$.
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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