TITLE: One polynomial problem
QUESTION [7 upvotes]: It's known that a polynomial $f\in\mathbb{C}[x]$, whose degree is $n$, possesses integer values at each of the points: $0,1,4,\ldots,n^2$. Prove that this polynomial possesses an integer value at $m^2$ for any $m$.
Thanks in advance!

REPLY [7 votes]: The polynomial $g(x) = f(x^2)$ takes integer values at $0,\pm 1,\ldots,\pm n$, so it suffices to show that $g(x-n)$ (which has degree $2n$ and integer values at $0,\ldots,2n$) takes an integer value at $m$ for any $m$. This is easy to see by writing $g(x-n)$ in terms of the basis $\binom{x}{0},\binom{x}{1},\binom{x}{2},\ldots$ for the space of polynomials.