Coefficients of a polynomial function from $\Bbb{N}$ to $\Bbb{N}$ Let $f$ be a polynomial from $\Bbb{N}$ to $\Bbb{N}$.
So let $f(x) = a_0+a_1x+a_2x^2+..a_rx^r$
Then for every natural number $k$, $f(k) \in \Bbb{N}$.
But how does it imply that all the coefficients of $f(x)$ i.e. $a_0, a_1,..a_r$ will belong to $\Bbb{Q}$?

You can retrieve the coefficients by using a Lagrangian interpolator on $r+1$ natural points. They will be rational.
The reasoning also applies to polynomials from $\mathbb Q$ to $\mathbb Q$.
